{"id":2869,"date":"2026-10-30T17:40:31","date_gmt":"2026-10-30T10:40:31","guid":{"rendered":"https:\/\/navi.edu.vn\/blog\/?p=2869"},"modified":"2026-10-30T17:40:31","modified_gmt":"2026-10-30T10:40:31","slug":"de-cuong-on-tap-giua-ki-1-toan-11","status":"publish","type":"post","link":"https:\/\/navi.edu.vn\/blog\/de-cuong-on-tap-giua-ki-1-toan-11\/","title":{"rendered":"\u0110\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 (2026-2027) &#038; \u0110\u1ec1 thi m\u1eabu c\u00f3 \u0111\u00e1p \u00e1n"},"content":{"rendered":"<div class=\"naviedu-intro-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<p>L\u1edbp 11 l\u00e0 n\u0103m h\u1ecdc b\u1ea3n l\u1ec1 mang t\u00ednh quy\u1ebft \u0111\u1ecbnh tr\u01b0\u1edbc khi b\u01b0\u1edbc v\u00e0o k\u1ef3 thi THPT qu\u1ed1c gia v\u00e0 x\u00e9t tuy\u1ec3n \u0111\u1ea1i h\u1ecdc. Giai \u0111o\u1ea1n gi\u1eefa k\u00ec 1, h\u1ecdc sinh ph\u1ea3i \u0111\u1ed1i m\u1eb7t v\u1edbi l\u01b0\u1ee3ng ki\u1ebfn th\u1ee9c n\u1eb7ng v\u00e0 tr\u1eebu t\u01b0\u1ee3ng g\u1ed3m chuy\u00ean \u0111\u1ec1 l\u01b0\u1ee3ng gi\u00e1c v\u1edbi h\u1ec7 th\u1ed1ng c\u00f4ng th\u1ee9c bi\u1ebfn \u0111\u1ed5i d\u1ec5 nh\u1ea7m l\u1eabn v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian ba chi\u1ec1u l\u1ea7n \u0111\u1ea7u ti\u1ebfp c\u1eadn \u0111\u1ea7y th\u1eed th\u00e1ch. Th\u00eam v\u00e0o \u0111\u00f3, \u0111\u1ecbnh d\u1ea1ng \u0111\u1ec1 ki\u1ec3m tra m\u1edbi v\u1edbi c\u00e1c c\u00e2u h\u1ecfi \u0111\u00fang\/sai v\u00e0 tr\u1ea3 l\u1eddi ng\u1eafn \u0111\u00f2i h\u1ecfi t\u01b0 duy logic s\u1eafc b\u00e9n.<\/p>\n<p>Hi\u1ec3u \u0111\u01b0\u1ee3c nh\u1eefng lo l\u1eafng \u0111\u00f3, NaviEdu bi\u00ean so\u1ea1n tr\u1ecdn b\u1ed9 <strong>\u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 to\u00e1n 11<\/strong> chu\u1ea9n h\u00f3a: h\u1ec7 th\u1ed1ng to\u00e0n b\u1ed9 l\u00fd thuy\u1ebft c\u1ed1t l\u00f5i, ph\u01b0\u01a1ng ph\u00e1p gi\u1ea3i chi ti\u1ebft, l\u1ecbch \u00f4n t\u1eadp khoa h\u1ecdc k\u00e8m \u0111\u1ec1 thi m\u1eabu c\u00f3 \u0111\u00e1p \u00e1n gi\u00fap h\u1ecdc sinh b\u1ee9t ph\u00e1 \u0111i\u1ec3m s\u1ed1.<\/p>\n<\/div>\n<figure style=\"margin: 24px 0; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2870\" style=\"max-width: 100%; height: auto; border-radius: 10px;\" src=\"https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-tom-tat.png\" alt=\"\u0110\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 \u2013 t\u00f3m t\u1eaft ki\u1ebfn th\u1ee9c tr\u1ecdng t\u00e2m\" width=\"1200\" height=\"760\" srcset=\"https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-tom-tat.png 1200w, https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-tom-tat-300x190.png 300w, https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-tom-tat-1024x649.png 1024w, https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-tom-tat-768x486.png 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><figcaption style=\"font-size: 14px; color: #718096; margin-top: 8px;\">S\u01a1 \u0111\u1ed3 t\u00f3m t\u1eaft \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 (2026\u20132027)<\/figcaption><\/figure>\n<div class=\"naviedu-toc\" style=\"background: #f7fafc; border: 1px solid #e2e8f0; border-radius: 8px; padding: 16px 22px; margin: 24px 0;\">\n<p style=\"margin: 0 0 8px 0; font-weight: bold; color: #0056b3; font-size: 17px;\">M\u1ee5c l\u1ee5c \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11<\/p>\n<ul style=\"margin: 0; padding-left: 22px; line-height: 1.9;\">\n<li style=\"font-weight: bold; list-style-type: disc;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-1-cau-truc-de-thi-giua-ki-1-toan-11-theo-dinh-dang-moi\">C\u1ea5u tr\u00fac \u0111\u1ec1 thi gi\u1eefa k\u00ec 1 To\u00e1n 11 theo \u0111\u1ecbnh d\u1ea1ng m\u1edbi<\/a><\/li>\n<li style=\"font-weight: bold; list-style-type: disc;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-2-phan-i-luong-giac\">Ph\u1ea7n I: L\u01b0\u1ee3ng Gi\u00e1c<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-3-cong-thuc-luong-giac-co-ban\">C\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-4-cong-thuc-cong-va-cong-thuc-nhan-doi\">C\u00f4ng th\u1ee9c c\u1ed9ng v\u00e0 c\u00f4ng th\u1ee9c nh\u00e2n \u0111\u00f4i<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-5-cong-thuc-bien-doi-tich-thanh-tong-va-tong-thanh-tich\">C\u00f4ng th\u1ee9c bi\u1ebfn \u0111\u1ed5i t\u00edch th\u00e0nh t\u1ed5ng v\u00e0 t\u1ed5ng th\u00e0nh t\u00edch<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-6-ham-so-luong-giac\">H\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-7-tap-xac-dinh-ham-so-luong-giac\">T\u1eadp x\u00e1c \u0111\u1ecbnh h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-8-tinh-tuan-hoan-cua-ham-so-luong-giac\">T\u00ednh tu\u1ea7n ho\u00e0n c\u1ee7a h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-9-phuong-trinh-luong-giac-co-ban\">Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-10-gia-tri-lon-nhat-nho-nhat-cua-bieu-thuc-luong-giac\">Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t, nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c<\/a><\/li>\n<li style=\"font-weight: bold; list-style-type: disc;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-11-phan-ii-duong-thang-va-mat-phang-trong-khong-gian\">Ph\u1ea7n II: \u0110\u01b0\u1eddng Th\u1eb3ng v\u00e0 M\u1eb7t Ph\u1eb3ng Trong Kh\u00f4ng Gian<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-12-duong-thang-va-mat-phang-trong-khong-gian\">\u0110\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng trong kh\u00f4ng gian<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-13-tim-giao-tuyen-cua-hai-mat-phang\">T\u00ecm giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-14-tim-giao-diem-cua-duong-thang-va-mat-phang\">T\u00ecm giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-15-phep-chieu-song-song\">Ph\u00e9p chi\u1ebfu song song<\/a><\/li>\n<li style=\"font-weight: bold; list-style-type: disc;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-16-phan-iii-lich-on-tap-7-ngay-de-thi-mau\">Ph\u1ea7n III: L\u1ecbch \u00d4n T\u1eadp 7 Ng\u00e0y &amp; \u0110\u1ec1 Thi M\u1eabu<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-17-lich-on-tap-giua-ki-1-toan-11-trong-7-ngay-giup-but-pha-diem\">L\u1ecbch \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 trong 7 ng\u00e0y gi\u00fap b\u1ee9t ph\u00e1 \u0111i\u1ec3m s\u1ed1<\/a><\/li>\n<li style=\"font-weight: normal; margin-left: 14px; list-style-type: circle;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-18-tai-de-cuong-on-tap-giua-ki-1-toan-11-va-de-thi-mau-pdf-co-d\">T\u1ea3i \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 v\u00e0 \u0111\u1ec1 thi m\u1eabu (PDF c\u00f3 \u0111\u00e1p \u00e1n)<\/a><\/li>\n<li style=\"font-weight: bold; list-style-type: disc;\"><a style=\"color: #2d3748; text-decoration: none;\" href=\"#muc-19-loi-ket\">L\u1eddi k\u1ebft<\/a><\/li>\n<\/ul>\n<\/div>\n<p><!-- B\u1eaeT \u0110\u1ea6U: C\u1ea4U TR\u00daC \u0110\u1ec0 THI GI\u1eeeA K\u00cc 1 TO\u00c1N 11 --><\/p>\n<div class=\"naviedu-content-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<h2 id=\"muc-1-cau-truc-de-thi-giua-ki-1-toan-11-theo-dinh-dang-moi\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 30px; font-size: 22px;\">C\u1ea5u tr\u00fac \u0111\u1ec1 thi gi\u1eefa k\u00ec 1 To\u00e1n 11 theo \u0111\u1ecbnh d\u1ea1ng m\u1edbi<\/h2>\n<figure id=\"attachment_2913\" aria-describedby=\"caption-attachment-2913\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2913\" src=\"https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-300x188.png\" alt=\"T\u00ecm hi\u1ec3u c\u1ea5u tr\u00fac m\u00e0 tr\u1eadn \u0111\u1ec1 thi gi\u1eefa k\u1ef3 1 To\u00e1n 11\" width=\"800\" height=\"500\" srcset=\"https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-300x188.png 300w, https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-1024x640.png 1024w, https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-768x480.png 768w, https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11-1536x961.png 1536w, https:\/\/navi.edu.vn\/blog\/wp-content\/uploads\/2026\/10\/de-cuong-on-tap-giua-ki-1-toan-11.png 1586w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><figcaption id=\"caption-attachment-2913\" class=\"wp-caption-text\">T\u00ecm hi\u1ec3u c\u1ea5u tr\u00fac m\u00e0 tr\u1eadn \u0111\u1ec1 thi gi\u1eefa k\u1ef3 1 To\u00e1n 11<\/figcaption><\/figure>\n<p>H\u1ecdc sinh c\u1ea5p THPT \u0111\u01b0\u1ee3c \u0111\u00e1nh gi\u00e1 k\u1ebft qu\u1ea3 h\u1ecdc t\u1eadp theo <a href=\"https:\/\/luatvietnam.vn\/giao-duc\/thong-tu-22-2021-tt-bgddt-207846-d1.html\" target=\"_blank\" rel=\"noopener\">Th\u00f4ng t\u01b0 22\/2021\/TT-BGD\u0110T<\/a>. \u0110\u1ec1 ki\u1ec3m tra \u0111\u1ecbnh k\u00ec m\u00f4n To\u00e1n l\u1edbp 11 theo \u0111\u1ecbnh d\u1ea1ng m\u1edbi \u00e1p d\u1ee5ng c\u1ea5u tr\u00fac 4 ph\u1ea7n nh\u1eb1m \u0111\u00e1nh gi\u00e1 to\u00e0n di\u1ec7n n\u0103ng l\u1ef1c t\u01b0 duy, kh\u1ea3 n\u0103ng bi\u1ebfn \u0111\u1ed5i v\u00e0 t\u01b0 duy kh\u00f4ng gian c\u1ee7a h\u1ecdc sinh. V\u00ec ma tr\u1eadn v\u00e0 \u0111\u1ec1 ki\u1ec3m tra ch\u00ednh th\u1ee9c do S\u1edf Gi\u00e1o d\u1ee5c v\u00e0 \u0110\u00e0o t\u1ea1o ho\u1eb7c t\u1eebng tr\u01b0\u1eddng THPT x\u00e2y d\u1ef1ng, c\u1ea5u tr\u00fac d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1eabu tham kh\u1ea3o ph\u1ed5 bi\u1ebfn nh\u1ea5t v\u1edbi t\u1ed5ng th\u1eddi gian l\u00e0m b\u00e0i l\u00e0 <strong>90 ph\u00fat<\/strong>:<\/p>\n<ul style=\"padding-left: 20px;\">\n<li><strong>Ph\u1ea7n I \u2013 Tr\u1eafc nghi\u1ec7m nhi\u1ec1u l\u1ef1a ch\u1ecdn (3,0 \u0111i\u1ec3m):<\/strong> G\u1ed3m 12 c\u00e2u h\u1ecfi (m\u1ed7i c\u00e2u 0,25 \u0111i\u1ec3m), y\u00eau c\u1ea7u h\u1ecdc sinh ch\u1ecdn 1 ph\u01b0\u01a1ng \u00e1n \u0111\u00fang duy nh\u1ea5t trong 4 l\u1ef1a ch\u1ecdn, ki\u1ec3m tra nhanh ki\u1ebfn th\u1ee9c nh\u1eadn bi\u1ebft v\u00e0 th\u00f4ng hi\u1ec3u v\u1ec1 g\u00f3c l\u01b0\u1ee3ng gi\u00e1c, c\u00e1c c\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c, h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c, ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian c\u01a1 b\u1ea3n.<\/li>\n<li><strong>Ph\u1ea7n II \u2013 Tr\u1eafc nghi\u1ec7m \u0111\u00fang\/sai (2,0 \u0111i\u1ec3m):<\/strong> G\u1ed3m 2 c\u00e2u h\u1ecfi l\u1edbn, m\u1ed7i c\u00e2u c\u00f3 4 \u00fd a, b, c, d (t\u1ed1i \u0111a 1 \u0111i\u1ec3m\/c\u00e2u). \u0110i\u1ec3m s\u1ed1 \u0111\u01b0\u1ee3c ch\u1ea5m l\u0169y ti\u1ebfn theo s\u1ed1 \u00fd l\u00e0m \u0111\u00fang trong m\u1ed7i c\u00e2u: \u0111\u00fang 1 \u00fd \u0111\u1ea1t 0,1 \u0111i\u1ec3m; \u0111\u00fang 2 \u00fd \u0111\u1ea1t 0,25 \u0111i\u1ec3m; \u0111\u00fang 3 \u00fd \u0111\u1ea1t 0,5 \u0111i\u1ec3m; \u0111\u00fang c\u1ea3 4 \u00fd \u0111\u1ea1t tr\u1ecdn v\u1eb9n 1,0 \u0111i\u1ec3m. Ph\u1ea7n n\u00e0y \u0111\u00f2i h\u1ecfi t\u01b0 duy logic s\u00e2u v\u00e0 kh\u1ea3 n\u0103ng ki\u1ec3m ch\u1ee9ng ch\u1eb7t ch\u1ebd c\u00e1c m\u1ec7nh \u0111\u1ec1 l\u01b0\u1ee3ng gi\u00e1c v\u00e0 quan h\u1ec7 kh\u00f4ng gian.<\/li>\n<li><strong>Ph\u1ea7n III \u2013 Tr\u1eafc nghi\u1ec7m tr\u1ea3 l\u1eddi ng\u1eafn (2,0 \u0111i\u1ec3m):<\/strong> G\u1ed3m 4 c\u00e2u h\u1ecfi (m\u1ed7i c\u00e2u 0,5 \u0111i\u1ec3m), y\u00eau c\u1ea7u h\u1ecdc sinh t\u1ef1 t\u00ednh to\u00e1n ra k\u1ebft qu\u1ea3 v\u00e0 \u0111i\u1ec1n v\u00e0o phi\u1ebfu tr\u1ea3 l\u1eddi, th\u1eed th\u00e1ch k\u1ef9 n\u0103ng bi\u1ebfn \u0111\u1ed5i c\u00f4ng th\u1ee9c v\u00e0 x\u00e1c \u0111\u1ecbnh giao tuy\u1ebfn, giao \u0111i\u1ec3m trong kh\u00f4ng gian.<\/li>\n<li><strong>Ph\u1ea7n IV \u2013 T\u1ef1 lu\u1eadn (3,0 \u0111i\u1ec3m):<\/strong> G\u1ed3m c\u00e1c b\u00e0i to\u00e1n t\u1ef1 lu\u1eadn \u0111\u00f2i h\u1ecfi k\u1ef9 n\u0103ng tr\u00ecnh b\u00e0y l\u1eddi gi\u1ea3i chi ti\u1ebft, r\u00f5 r\u00e0ng \u0111\u1ed1i v\u1edbi gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c, t\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t nh\u1ecf nh\u1ea5t v\u00e0 b\u00e0i to\u00e1n ch\u1ee9ng minh h\u00ecnh h\u1ecdc kh\u00f4ng gian (t\u00ecm giao tuy\u1ebfn, giao \u0111i\u1ec3m).<\/li>\n<\/ul>\n<h3 style=\"color: #2b6cb0; font-size: 18px; margin-top: 22px;\">B\u1ea3ng ma tr\u1eadn tham kh\u1ea3o ph\u00e2n b\u1ed5 n\u1ed9i dung v\u00e0 h\u00ecnh th\u1ee9c<\/h3>\n<div style=\"overflow-x: auto; margin: 15px 0;\">\n<table style=\"min-width: 640px; table-layout: fixed; font-size: 15px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: left;\">\n<colgroup>\n<col style=\"width: 25%;\" \/>\n<col style=\"width: 15%;\" \/>\n<col style=\"width: 15%;\" \/>\n<col style=\"width: 15%;\" \/>\n<col style=\"width: 15%;\" \/>\n<col style=\"width: 15%;\" \/><\/colgroup>\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; color: #0056b3;\">Ch\u1ee7 \u0111\u1ec1<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; color: #0056b3; text-align: center;\">Nhi\u1ec1u l\u1ef1a ch\u1ecdn<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; color: #0056b3; text-align: center;\">\u0110\u00fang \/ Sai<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; color: #0056b3; text-align: center;\">Tr\u1ea3 l\u1eddi ng\u1eafn<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; color: #0056b3; text-align: center;\">T\u1ef1 lu\u1eadn<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; color: #0056b3; text-align: center;\">T\u1ed5ng \u0111i\u1ec3m<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; font-weight: bold;\">C\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c v\u00e0 h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">1,5 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">1,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">0,5 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">0,5 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center; font-weight: bold;\">3,5 \u0111i\u1ec3m<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; font-weight: bold;\">Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c, GTLN \u2013 GTNN<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">0,75 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">1,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">1,5 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center; font-weight: bold;\">3,25 \u0111i\u1ec3m<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; font-weight: bold;\">\u0110\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng trong kh\u00f4ng gian<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">0,75 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">1,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">0,5 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">1,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center; font-weight: bold;\">3,25 \u0111i\u1ec3m<\/td>\n<\/tr>\n<tr style=\"background-color: #f7fafc; font-weight: bold;\">\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; color: #0056b3;\">T\u1ed4NG \u0110I\u1ec2M<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">3,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">2,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">2,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center;\">3,0 \u0111i\u1ec3m<\/td>\n<td style=\"white-space: nowrap; word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px 12px; border: 1px solid #cbd5e0; text-align: center; color: #c53030;\">10,0 \u0111i\u1ec3m<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><!-- H\u1ed8P L\u01afU \u00dd --><\/p>\n<div style=\"background-color: #edf2f7; border-left: 4px solid #0056b3; padding: 14px 18px; border-radius: 4px; margin-top: 25px;\">\n<p style=\"margin: 0; font-size: 15px; color: #2d3748;\"><strong>\ud83d\udca1 L\u01b0u \u00fd quan tr\u1ecdng cho h\u1ecdc sinh:<\/strong> C\u1ea5u tr\u00fac ma tr\u1eadn \u0111\u1ec1 thi c\u00f3 th\u1ec3 c\u00f3 nh\u1eefng \u0111i\u1ec1u ch\u1ec9nh linh ho\u1ea1t t\u00f9y theo ti\u1ebfn \u0111\u1ed9 gi\u1ea3ng d\u1ea1y v\u00e0 y\u00eau c\u1ea7u c\u1ee7a t\u1eebng tr\u01b0\u1eddng THPT (v\u00ed d\u1ee5: m\u1ed9t s\u1ed1 tr\u01b0\u1eddng c\u00f3 th\u1ec3 ch\u01b0a \u0111\u01b0a ph\u1ea7n ph\u00e9p chi\u1ebfu song song v\u00e0o \u0111\u1ec1 ki\u1ec3m tra gi\u1eefa k\u00ec 1). Do \u0111\u00f3, c\u00e1c em n\u00ean tham kh\u1ea3o k\u1ef9 \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp do gi\u00e1o vi\u00ean b\u1ed9 m\u00f4n t\u1ea1i tr\u01b0\u1eddng ph\u00e1t ra \u0111\u1ec3 n\u1eafm ch\u00ednh x\u00e1c ph\u1ea1m vi \u00f4n thi.<\/p>\n<\/div>\n<\/div>\n<p><!-- K\u1ebeT TH\u00daC: C\u1ea4U TR\u00daC \u0110\u1ec0 THI GI\u1eeeA K\u00cc 1 TO\u00c1N 11 --><\/p>\n<p><!-- B\u1eaeT \u0110\u1ea6U: PH\u1ea6N I - L\u01af\u1ee2NG GI\u00c1C TO\u00c1N 11 --><\/p>\n<div class=\"naviedu-content-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<p><!-- BANNER PH\u1ea6N I --><\/p>\n<div style=\"background: linear-gradient(135deg, #0056b3, #0080ff); color: #ffffff; padding: 18px 24px; border-radius: 8px; margin-top: 35px; margin-bottom: 30px;\">\n<h2 id=\"muc-2-phan-i-luong-giac\" style=\"color: #ffffff; margin: 0; font-size: 22px; text-transform: uppercase; letter-spacing: 0.5px;\">Ph\u1ea7n I: L\u01b0\u1ee3ng Gi\u00e1c<\/h2>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 1: C\u00d4NG TH\u1ee8C L\u01af\u1ee2NG GI\u00c1C C\u01a0 B\u1ea2N --><\/p>\n<p style=\"margin: 0 0 16px 0;\">L\u01b0\u1ee3ng gi\u00e1c l\u00e0 ph\u1ea7n chi\u1ebfm nhi\u1ec1u \u0111i\u1ec3m nh\u1ea5t trong <strong>\u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11<\/strong>. Con n\u00ean h\u1ecdc theo \u0111\u00fang th\u1ee9 t\u1ef1: c\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c tr\u01b0\u1edbc, r\u1ed3i \u0111\u1ebfn h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c v\u00e0 cu\u1ed1i c\u00f9ng l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh.<\/p>\n<h2 id=\"muc-3-cong-thuc-luong-giac-co-ban\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">C\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">G\u00f3c l\u01b0\u1ee3ng gi\u00e1c, radian v\u00e0 c\u00e1c h\u1eb1ng \u0111\u1eb3ng th\u1ee9c c\u01a1 b\u1ea3n<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>\u0110\u01a1n v\u1ecb Radian:<\/strong> G\u00f3c \u03b1 c\u00f3 s\u1ed1 \u0111o b\u1eb1ng rad khi cung t\u01b0\u01a1ng \u1ee9ng tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n l\u01b0\u1ee3ng gi\u00e1c c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng b\u00e1n k\u00ednh. C\u00f4ng th\u1ee9c \u0111\u1ed5i \u0111\u01a1n v\u1ecb: <code>\u03c0 rad = 180\u00b0<\/code>, suy ra <code>1 rad = 180\u00b0 \/ \u03c0<\/code> v\u00e0 <code>1\u00b0 = \u03c0 \/ 180 rad<\/code>.<\/li>\n<li><strong>C\u00e1c h\u1eb1ng \u0111\u1eb3ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n:<\/strong>\n<div style=\"background: #edf2f7; padding: 8px 12px; border-radius: 6px; margin: 6px 0; font-family: monospace; font-weight: bold; color: #2b6cb0;\">sin<sup>2<\/sup>\u03b1 + cos<sup>2<\/sup>\u03b1 = 1<br \/>\ntan \u03b1 \u00b7 cot \u03b1 = 1 (v\u1edbi \u0111i\u1ec1u ki\u1ec7n sin \u03b1 kh\u00e1c 0, cos \u03b1 kh\u00e1c 0)<br \/>\n1 + tan<sup>2<\/sup>\u03b1 = 1 \/ (cos<sup>2<\/sup>\u03b1) (v\u1edbi cos \u03b1 kh\u00e1c 0)<br \/>\n1 + cot<sup>2<\/sup>\u03b1 = 1 \/ (sin<sup>2<\/sup>\u03b1) (v\u1edbi sin \u03b1 kh\u00e1c 0)<\/div>\n<\/li>\n<\/ul>\n<p><!-- B\u1ea2NG CUNG LI\u00caN QUAN \u0110\u1eb6C BI\u1ec6T --><\/p>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">Gi\u00e1 tr\u1ecb l\u01b0\u1ee3ng gi\u00e1c c\u00e1c cung li\u00ean quan \u0111\u1eb7c bi\u1ec7t<\/h3>\n<div style=\"overflow-x: auto; margin: 15px 0;\">\n<table style=\"table-layout: auto; font-size: 14px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: center;\">\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; color: #0056b3;\">Lo\u1ea1i cung<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0;\">G\u00f3c li\u00ean quan<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0;\">C\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c t\u01b0\u01a1ng \u1ee9ng<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; font-weight: bold;\">Cung \u0111\u1ed1i nhau<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0;\">\u03b1 v\u00e0 -\u03b1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; text-align: left;\">cos(-\u03b1) = cos \u03b1<br \/>\nsin(-\u03b1) = -sin \u03b1<br \/>\ntan(-\u03b1) = -tan \u03b1 | cot(-\u03b1) = -cot \u03b1<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; font-weight: bold;\">Cung b\u00f9 nhau<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0;\">\u03b1 v\u00e0 \u03c0 &#8211; \u03b1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; text-align: left;\">sin(\u03c0 &#8211; \u03b1) = sin \u03b1<br \/>\ncos(\u03c0 &#8211; \u03b1) = -cos \u03b1<br \/>\ntan(\u03c0 &#8211; \u03b1) = -tan \u03b1 | cot(\u03c0 &#8211; \u03b1) = -cot \u03b1<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; font-weight: bold;\">Cung ph\u1ee5 nhau<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0;\">\u03b1 v\u00e0 \u03c0\/2 &#8211; \u03b1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; text-align: left;\">sin(\u03c0\/2 &#8211; \u03b1) = cos \u03b1 | cos(\u03c0\/2 &#8211; \u03b1) = sin \u03b1<br \/>\ntan(\u03c0\/2 &#8211; \u03b1) = cot \u03b1 | cot(\u03c0\/2 &#8211; \u03b1) = tan \u03b1<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; font-weight: bold;\">Cung h\u01a1n k\u00e9m \u03c0<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0;\">\u03b1 v\u00e0 \u03b1 + \u03c0<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 8px; border: 1px solid #cbd5e0; text-align: left;\">sin(\u03b1 + \u03c0) = -sin \u03b1 | cos(\u03b1 + \u03c0) = -cos \u03b1<br \/>\ntan(\u03b1 + \u03c0) = tan \u03b1 | cot(\u03b1 + \u03c0) = cot \u03b1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p \u1edf c\u00f4ng th\u1ee9c c\u01a1 b\u1ea3n:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Nh\u1ea7m l\u1eabn d\u1ea5u tr\u1eeb (-) khi bi\u1ebfn \u0111\u1ed5i c\u00e1c cung b\u00f9 nhau ho\u1eb7c cung \u0111\u1ed1i (v\u00ed d\u1ee5 ng\u1ed9 nh\u1eadn cos(-\u03b1) b\u1eb1ng -cos \u03b1, ho\u1eb7c qu\u00ean \u0111\u1ed5i d\u1ea5u tan \u1edf cung b\u00f9).<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU (D\u1ea0NG \u0110\u00daNG\/SAI A, B, C, D) --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: X\u00e9t t\u00ednh \u0111\u00fang sai v\u1ec1 cung li\u00ean quan \u0111\u1eb7c bi\u1ec7t<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> Cho g\u00f3c \u03b1 t\u00f9y \u00fd. X\u00e9t t\u00ednh \u0111\u00fang sai c\u1ee7a c\u00e1c ph\u00e1t bi\u1ec3u sau:<br \/>\na) Bi\u1ec3u th\u1ee9c sin(-\u03b1) + sin(\u03c0 + \u03b1) lu\u00f4n b\u1eb1ng 0 v\u1edbi m\u1ecdi \u03b1.<br \/>\nb) cos(\u03c0 &#8211; \u03b1) b\u1eb1ng cos \u03b1.<br \/>\nc) tan(\u03c0\/2 &#8211; \u03b1) nh\u00e2n tan \u03b1 b\u1eb1ng 1.<br \/>\nd) N\u1ebfu sin \u03b1 b\u1eb1ng 3\/5 (v\u1edbi 0 &lt; \u03b1 &lt; \u03c0\/2) th\u00ec cos \u03b1 b\u1eb1ng 4\/5.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 <strong>\u00dd a:<\/strong> Ta c\u00f3 sin(-\u03b1) = -sin \u03b1 v\u00e0 sin(\u03c0 + \u03b1) = -sin \u03b1. T\u1ed5ng c\u1ee7a ch\u00fang b\u1eb1ng -2 sin \u03b1 (kh\u00f4ng \u0111\u1ed3ng nh\u1ea5t b\u1eb1ng 0). Ph\u00e1t bi\u1ec3u n\u00e0y <strong>sai<\/strong>.<br \/>\n\u2022 <strong>\u00dd b:<\/strong> Theo cung b\u00f9 nhau, cos(\u03c0 &#8211; \u03b1) = -cos \u03b1. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>sai<\/strong>.<br \/>\n\u2022 <strong>\u00dd c:<\/strong> Theo cung ph\u1ee5 nhau, tan(\u03c0\/2 &#8211; \u03b1) = cot \u03b1.<\/p>\n<p style=\"margin: 0;\">Do \u0111\u00f3 cot \u03b1 \u00b7 tan \u03b1 = 1. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>\u0111\u00fang<\/strong>.<br \/>\n\u2022 <strong>\u00dd d:<\/strong> V\u00ec sin<sup>2<\/sup>\u03b1 + cos<sup>2<\/sup>\u03b1 = 1 n\u00ean cos<sup>2<\/sup>\u03b1 = 1 &#8211; (3\/5)<sup>2<\/sup> = 16\/25. V\u00ec g\u00f3c \u03b1 thu\u1ed9c g\u00f3c ph\u1ea7n ph\u1ea7n t\u01b0 th\u1ee9 nh\u1ea5t (0 &lt; \u03b1 &lt; \u03c0\/2) n\u00ean cos \u03b1 d\u01b0\u01a1ng, suy ra cos \u03b1 = 4\/5. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>\u0111\u00fang<\/strong>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 2: C\u00d4NG TH\u1ee8C C\u1ed8NG V\u00c0 NH\u00c2N \u0110\u00d4I --><\/p>\n<h2 id=\"muc-4-cong-thuc-cong-va-cong-thuc-nhan-doi\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">C\u00f4ng th\u1ee9c c\u1ed9ng v\u00e0 c\u00f4ng th\u1ee9c nh\u00e2n \u0111\u00f4i<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">H\u1ec7 th\u1ed1ng c\u00f4ng th\u1ee9c c\u1ed9ng v\u00e0 nh\u00e2n \u0111\u00f4i<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>C\u00f4ng th\u1ee9c c\u1ed9ng l\u01b0\u1ee3ng gi\u00e1c:<\/strong>\n<div style=\"background: #edf2f7; padding: 8px 12px; border-radius: 6px; margin: 6px 0; font-family: monospace; font-weight: bold; color: #2b6cb0;\">cos(a \u00b1 b) = cos a cos b \u2213 sin a sin b<br \/>\nsin(a \u00b1 b) = sin a cos b \u00b1 cos a sin b<br \/>\ntan(a \u00b1 b) = (tan a \u00b1 tan b) \/ (1 \u2213 tan a tan b)<\/div>\n<\/li>\n<li><strong>C\u00f4ng th\u1ee9c nh\u00e2n \u0111\u00f4i v\u00e0 h\u1ea1 b\u1eadc:<\/strong>\n<div style=\"background: #edf2f7; padding: 8px 12px; border-radius: 6px; margin: 6px 0; font-family: monospace; font-weight: bold; color: #2b6cb0;\">sin 2a = 2 sin a cos a<br \/>\ncos 2a = cos<sup>2<\/sup>a &#8211; sin<sup>2<\/sup>a = 2 cos<sup>2<\/sup>a &#8211; 1 = 1 &#8211; 2 sin<sup>2<\/sup>a<br \/>\ntan 2a = (2 tan a) \/ (1 &#8211; tan<sup>2<\/sup>a)<br \/>\nC\u00f4ng th\u1ee9c h\u1ea1 b\u1eadc: cos<sup>2<\/sup>a = (1 + cos 2a) \/ 2 \u00a0\u00a0|\u00a0\u00a0 sin<sup>2<\/sup>a = (1 &#8211; cos 2a) \/ 2<\/div>\n<\/li>\n<\/ul>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p \u1edf c\u00f4ng th\u1ee9c c\u1ed9ng v\u00e0 nh\u00e2n \u0111\u00f4i:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Nh\u1ea7m l\u1eabn d\u1ea5u trong c\u00f4ng th\u1ee9c cos(a + b) (vi\u1ebft th\u00e0nh d\u1ea5u c\u1ed9ng thay v\u00ec d\u1ea5u tr\u1eeb) ho\u1eb7c qu\u00ean h\u1ec7 s\u1ed1 2 trong c\u00f4ng th\u1ee9c sin 2a v\u00e0 c\u00f4ng th\u1ee9c h\u1ea1 b\u1eadc.<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: R\u00fat g\u1ecdn bi\u1ec3u th\u1ee9c b\u1eb1ng c\u00f4ng th\u1ee9c nh\u00e2n \u0111\u00f4i<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> R\u00fat g\u1ecdn bi\u1ec3u th\u1ee9c sau: <code>A = (sin 2a) \/ (sin a) - (cos 2a) \/ (cos a)<\/code> (v\u1edbi \u0111i\u1ec1u ki\u1ec7n bi\u1ec3u th\u1ee9c c\u00f3 ngh\u0129a).<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c nh\u00e2n \u0111\u00f4i cho sin 2a v\u00e0 m\u1ed9t d\u1ea1ng c\u1ee7a cos 2a:<br \/>\n&#8211; <code>sin 2a = 2 sin a cos a<\/code><br \/>\n&#8211; <code>cos 2a = 2 cos<sup>2<\/sup>a - 1<\/code> (ho\u1eb7c d\u00f9ng <code>cos<sup>2<\/sup>a - sin<sup>2<\/sup>a<\/code>)<br \/>\n\u2022 Thay v\u00e0o bi\u1ec3u th\u1ee9c A:<\/p>\n<p style=\"margin: 0;\"><code>A = (2 sin a cos a) \/ (sin a) - (2 cos<sup>2<\/sup>a - 1) \/ (cos a)<\/code><br \/>\n<code>A = 2 cos a - (2 cos a - 1 \/ cos a) = 2 cos a - 2 cos a + 1 \/ cos a = <strong>1 \/ cos a<\/strong><\/code>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 3: C\u00d4NG TH\u1ee8C BI\u1ebeN \u0110\u1ed4I T\u00cdCH TH\u00c0NH T\u1ed4NG V\u00c0 T\u1ed4NG TH\u00c0NH T\u00cdCH --><\/p>\n<h2 id=\"muc-5-cong-thuc-bien-doi-tich-thanh-tong-va-tong-thanh-tich\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">C\u00f4ng th\u1ee9c bi\u1ebfn \u0111\u1ed5i t\u00edch th\u00e0nh t\u1ed5ng v\u00e0 t\u1ed5ng th\u00e0nh t\u00edch<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">H\u1ec7 th\u1ed1ng c\u00f4ng th\u1ee9c bi\u1ebfn \u0111\u1ed5i<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>C\u00f4ng th\u1ee9c bi\u1ebfn \u0111\u1ed5i t\u00edch th\u00e0nh t\u1ed5ng:<\/strong>\n<div style=\"background: #edf2f7; padding: 8px 12px; border-radius: 6px; margin: 6px 0; font-family: monospace; font-weight: bold; color: #2b6cb0;\">cos a cos b = \u00bd [cos(a &#8211; b) + cos(a + b)]<br \/>\nsin a sin b = \u00bd [cos(a &#8211; b) &#8211; cos(a + b)]<br \/>\nsin a cos b = \u00bd [sin(a &#8211; b) + sin(a + b)]<\/div>\n<\/li>\n<li><strong>C\u00f4ng th\u1ee9c bi\u1ebfn \u0111\u1ed5i t\u1ed5ng th\u00e0nh t\u00edch:<\/strong>\n<div style=\"background: #edf2f7; padding: 8px 12px; border-radius: 6px; margin: 6px 0; font-family: monospace; font-weight: bold; color: #2b6cb0;\">cos p + cos q = 2 cos((p + q) \/ 2) cos((p &#8211; q) \/ 2)<br \/>\ncos p &#8211; cos q = -2 sin((p + q) \/ 2) sin((p &#8211; q) \/ 2)<br \/>\nsin p + sin q = 2 sin((p + q) \/ 2) cos((p &#8211; q) \/ 2)<br \/>\nsin p &#8211; sin q = 2 cos((p + q) \/ 2) sin((p &#8211; q) \/ 2)<\/div>\n<\/li>\n<\/ul>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p khi bi\u1ebfn \u0111\u1ed5i t\u1ed5ng th\u00e0nh t\u00edch:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Qu\u00ean h\u1ec7 s\u1ed1 2 ho\u1eb7c qu\u00ean chia \u0111\u00f4i c\u00e1c t\u1ed5ng\/hi\u1ec7u g\u00f3c p v\u00e0 q (v\u00ed d\u1ee5 vi\u1ebft th\u00e0nh cos p + cos q = 2 cos(p + q) cos(p &#8211; q) m\u00e0 thi\u1ebfu chia 2 \u1edf g\u00f3c).<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: Bi\u1ec3u di\u1ec5n t\u1ed5ng th\u00e0nh t\u00edch \u0111\u1ec3 r\u00fat g\u1ecdn<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> R\u00fat g\u1ecdn bi\u1ec3u th\u1ee9c: <code>S = (sin 5x + sin x) \/ (cos 5x + cos x)<\/code>.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c t\u1ed5ng th\u00e0nh t\u00edch cho t\u1eed s\u1ed1 v\u00e0 m\u1eabu s\u1ed1:<br \/>\n&#8211; T\u1eed s\u1ed1: <code>sin 5x + sin x = 2 sin((5x + x) \/ 2) cos((5x - x) \/ 2) = 2 sin 3x cos 2x<\/code>.<br \/>\n&#8211; M\u1eabu s\u1ed1:<\/p>\n<p style=\"margin: 0;\"><code>cos 5x + cos x = 2 cos((5x + x) \/ 2) cos((5x - x) \/ 2) = 2 cos 3x cos 2x<\/code>.<br \/>\n\u2022 Thay v\u00e0o bi\u1ec3u th\u1ee9c S:<br \/>\n<code>S = (2 sin 3x cos 2x) \/ (2 cos 3x cos 2x) = <strong>tan 3x<\/strong><\/code> (v\u1edbi \u0111i\u1ec1u ki\u1ec7n m\u1eabu s\u1ed1 kh\u00e1c 0).<\/p>\n<\/div>\n<\/div>\n<p><!-- K\u1ebeT TH\u00daC: PH\u1ea6N I - L\u01af\u1ee2NG GI\u00c1C TO\u00c1N 11 --><\/p>\n<p><!-- B\u1eaeT \u0110\u1ea6U: PH\u1ea6N I (TI\u1ebeP THEO) - H\u00c0M S\u1ed0 L\u01af\u1ee2NG GI\u00c1C TO\u00c1N 11 --><\/p>\n<div class=\"naviedu-content-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<p><!-- CH\u1ee6 \u0110\u1ec0 4: H\u00c0M S\u1ed0 L\u01af\u1ee2NG GI\u00c1C --><\/p>\n<p style=\"margin: 0 0 16px 0;\">Sau khi n\u1eafm ch\u1eafc c\u00f4ng th\u1ee9c, \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 chuy\u1ec3n sang h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c: t\u1eadp x\u00e1c \u0111\u1ecbnh, t\u00ednh ch\u1eb5n l\u1ebb v\u00e0 chu k\u00ec, nh\u1eefng n\u1ed9i dung r\u1ea5t hay xu\u1ea5t hi\u1ec7n \u1edf c\u00e2u \u0111\u00fang\/sai.<\/p>\n<h2 id=\"muc-6-ham-so-luong-giac\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">H\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">T\u1ed5ng h\u1ee3p t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n c\u1ee7a b\u1ed1n h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/h3>\n<p>Ch\u01b0\u01a1ng tr\u00ecnh To\u00e1n 11 gi\u1edbi thi\u1ec7u b\u1ed1n h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n bao g\u1ed3m h\u00e0m s\u1ed1 sin, h\u00e0m s\u1ed1 c\u00f4sin, h\u00e0m s\u1ed1 tang v\u00e0 h\u00e0m s\u1ed1 c\u00f4tang. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 b\u1ea3ng t\u00f3m t\u1eaft chi ti\u1ebft t\u1eadp gi\u00e1 tr\u1ecb, t\u00ednh ch\u1eb5n l\u1ebb v\u00e0 t\u00ednh ch\u1ea5t bi\u1ebfn thi\u00ean c\u1ee7a t\u1eebng h\u00e0m s\u1ed1:<\/p>\n<p><!-- B\u1ea2NG T\u00d3M T\u1eaeT T\u00cdNH CH\u1ea4T H\u00c0M S\u1ed0 L\u01af\u1ee2NG GI\u00c1C --><\/p>\n<div style=\"overflow-x: auto; margin: 15px 0;\">\n<table style=\"min-width: 640px; table-layout: fixed; font-size: 14px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: left;\">\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; color: #0056b3;\">H\u00e0m s\u1ed1<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">T\u1eadp x\u00e1c \u0111\u1ecbnh<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">T\u1eadp gi\u00e1 tr\u1ecb<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">T\u00ednh ch\u1eb5n l\u1ebb<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">T\u00ednh bi\u1ebfn thi\u00ean c\u01a1 b\u1ea3n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = sin x<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">R<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">[-1; 1]<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">H\u00e0m s\u1ed1 l\u1ebb<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">\u0110\u1ed3ng bi\u1ebfn tr\u00ean m\u1ed7i kho\u1ea3ng (-\u03c0\/2 + k2\u03c0; \u03c0\/2 + k2\u03c0)<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = cos x<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">R<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">[-1; 1]<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">H\u00e0m s\u1ed1 ch\u1eb5n<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">\u0110\u1ed3ng bi\u1ebfn tr\u00ean m\u1ed7i kho\u1ea3ng (-\u03c0 + k2\u03c0; k2\u03c0)<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = tan x<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">R \\ {\u03c0\/2 + k\u03c0, k \u2208 Z}<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">R<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">H\u00e0m s\u1ed1 l\u1ebb<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">\u0110\u1ed3ng bi\u1ebfn tr\u00ean m\u1ed7i kho\u1ea3ng (-\u03c0\/2 + k\u03c0; \u03c0\/2 + k\u03c0)<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = cot x<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">R \\ {k\u03c0, k \u2208 Z}<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">R<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">H\u00e0m s\u1ed1 l\u1ebb<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">Ngh\u1ecbch bi\u1ebfn tr\u00ean m\u1ed7i kho\u1ea3ng (k\u03c0; \u03c0 + k\u03c0)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p v\u1ec1 t\u00ednh ch\u1eb5n l\u1ebb:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Nhi\u1ec1u h\u1ecdc sinh nh\u1edb nh\u1ea7m h\u00e0m s\u1ed1 y = cos x l\u00e0 h\u00e0m l\u1ebb ho\u1eb7c nh\u1ea7m l\u1eabn kho\u1ea3ng \u0111\u1ed3ng bi\u1ebfn, ngh\u1ecbch bi\u1ebfn c\u1ee7a h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c khi gi\u1ea3i c\u00e1c c\u00e2u h\u1ecfi tr\u1eafc nghi\u1ec7m \u0111\u00fang\/sai.<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU (D\u1ea0NG \u0110\u00daNG\/SAI A, B, C, D) --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: Kh\u1ea3o s\u00e1t t\u00ednh ch\u1ea5t h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> X\u00e9t t\u00ednh \u0111\u00fang sai c\u1ee7a c\u00e1c ph\u00e1t bi\u1ec3u sau \u0111\u00e2y v\u1ec1 h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c:<br \/>\na) H\u00e0m s\u1ed1 y = cos x l\u00e0 h\u00e0m s\u1ed1 ch\u1eb5n tr\u00ean t\u1eadp s\u1ed1 th\u1ef1c R.<br \/>\nb) T\u1eadp gi\u00e1 tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = sin x l\u00e0 \u0111o\u1ea1n t\u1eeb -1 \u0111\u1ebfn 1.<br \/>\nc) H\u00e0m s\u1ed1 y = tan x l\u00e0 h\u00e0m s\u1ed1 ch\u1eb5n.<br \/>\nd) \u0110\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = cot x nh\u1eadn g\u1ed1c t\u1ecda \u0111\u1ed9 l\u00e0m t\u00e2m \u0111\u1ed1i x\u1ee9ng.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 <strong>\u00dd a:<\/strong> Theo \u0111\u1ecbnh ngh\u0129a v\u00e0 b\u1ea3ng t\u00ednh ch\u1ea5t, cos(-x) = cos x n\u00ean h\u00e0m s\u1ed1 y = cos x l\u00e0 h\u00e0m ch\u1eb5n. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>\u0111\u00fang<\/strong>.<br \/>\n\u2022 <strong>\u00dd b:<\/strong> Gi\u00e1 tr\u1ecb c\u1ee7a sin x lu\u00f4n dao \u0111\u1ed9ng trong \u0111o\u1ea1n t\u1eeb -1 \u0111\u1ebfn 1.<\/p>\n<p style=\"margin: 0;\">Ph\u00e1t bi\u1ec3u n\u00e0y <strong>\u0111\u00fang<\/strong>.<br \/>\n\u2022 <strong>\u00dd c:<\/strong> Ta c\u00f3 tan(-x) = -tan x n\u00ean h\u00e0m s\u1ed1 y = tan x l\u00e0 h\u00e0m s\u1ed1 l\u1ebb, kh\u00f4ng ph\u1ea3i h\u00e0m ch\u1eb5n. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>sai<\/strong>.<br \/>\n\u2022 <strong>\u00dd d:<\/strong> V\u00ec y = cot x l\u00e0 h\u00e0m s\u1ed1 l\u1ebb n\u00ean \u0111\u1ed3 th\u1ecb c\u1ee7a n\u00f3 nh\u1eadn g\u1ed1c t\u1ecda \u0111\u1ed9 O(0; 0) l\u00e0m t\u00e2m \u0111\u1ed1i x\u1ee9ng. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>\u0111\u00fang<\/strong>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 5: T\u1eacP X\u00c1C \u0110\u1ecaNH H\u00c0M S\u1ed0 L\u01af\u1ee2NG GI\u00c1C --><\/p>\n<h2 id=\"muc-7-tap-xac-dinh-ham-so-luong-giac\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">T\u1eadp x\u00e1c \u0111\u1ecbnh h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">Ph\u01b0\u01a1ng ph\u00e1p t\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/h3>\n<p>Khi t\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c ch\u1ee9a ph\u00e2n th\u1ee9c ho\u1eb7c c\u0103n th\u1ee9c, h\u1ecdc sinh c\u1ea7n tu\u00e2n th\u1ee7 c\u00e1c \u0111i\u1ec1u ki\u1ec7n r\u00e0ng bu\u1ed9c to\u00e1n h\u1ecdc quen thu\u1ed9c:<\/p>\n<ul style=\"padding-left: 20px;\">\n<li>Bi\u1ec3u th\u1ee9c \u1edf m\u1eabu s\u1ed1 ph\u1ea3i kh\u00e1c 0 (v\u00ed d\u1ee5: h\u00e0m tan x c\u00f3 m\u1eabu cos x kh\u00e1c 0, h\u00e0m cot x c\u00f3 m\u1eabu sin x kh\u00e1c 0).<\/li>\n<li>Bi\u1ec3u th\u1ee9c d\u01b0\u1edbi d\u1ea5u c\u0103n b\u1eadc ch\u1eb5n ph\u1ea3i l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 0 (v\u00ed d\u1ee5: c\u0103n b\u1eadc hai c\u1ee7a A x\u00e1c \u0111\u1ecbnh khi A l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 0).<\/li>\n<li>H\u00e0m s\u1ed1 ch\u1ee9a ph\u00e2n th\u1ee9c d\u1ea1ng 1 \/ (sin x &#8211; a) x\u00e1c \u0111\u1ecbnh khi sin x kh\u00e1c a.<\/li>\n<\/ul>\n<p><!-- B\u1ea2NG T\u00d3M T\u1eaeT \u0110I\u1ec0U KI\u1ec6N X\u00c1C \u0110\u1ecaNH --><\/p>\n<div style=\"overflow-x: auto; margin: 15px 0;\">\n<table style=\"table-layout: auto; font-size: 14px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: left;\">\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; color: #0056b3;\">D\u1ea1ng h\u00e0m s\u1ed1<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">\u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">K\u1ebft qu\u1ea3 t\u1eadp x\u00e1c \u0111\u1ecbnh chu\u1ea9n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = 1 \/ (cos x)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">cos x kh\u00e1c 0<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">x kh\u00e1c \u03c0\/2 + k\u03c0, k \u2208 Z<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = 1 \/ (sin x)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">sin x kh\u00e1c 0<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">x kh\u00e1c k\u03c0, k \u2208 Z<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = 1 \/ (sin x &#8211; 1)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">sin x kh\u00e1c 1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">x kh\u00e1c \u03c0\/2 + k2\u03c0, k \u2208 Z<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = \u221a(1 &#8211; cos x)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">1 &#8211; cos x l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 0 (lu\u00f4n \u0111\u00fang)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">D = R<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p khi t\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Qu\u00ean \u0111i\u1ec1u ki\u1ec7n k thu\u1ed9c Z khi vi\u1ebft k\u1ebft qu\u1ea3 t\u1eadp x\u00e1c \u0111\u1ecbnh ho\u1eb7c gi\u1ea3i sai ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n khi cho m\u1eabu s\u1ed1 b\u1eb1ng 0 (v\u00ed d\u1ee5 gi\u1ea3i sin x = 1 nh\u01b0ng l\u1ea1i l\u1ea5y nh\u1ea7m h\u1ecd nghi\u1ec7m c\u1ee7a sin x = 0).<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 ch\u1ee9a ph\u00e2n th\u1ee9c<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh D c\u1ee7a h\u00e0m s\u1ed1: <code>y = 1 \/ (sin x - cos x)<\/code>.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 H\u00e0m s\u1ed1 x\u00e1c \u0111\u1ecbnh khi v\u00e0 ch\u1ec9 khi m\u1eabu s\u1ed1 kh\u00e1c 0:<br \/>\n<code>sin x - cos x kh\u00e1c 0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>sin x kh\u00e1c cos x<\/code>.<br \/>\n\u2022 Chia c\u1ea3 hai v\u1ebf cho cos x (v\u1edbi cos x kh\u00e1c 0), ta \u0111\u01b0\u1ee3c:<br \/>\n<code>tan x kh\u00e1c 1<\/code>.<br \/>\n\u2022 Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh <code>tan x = 1<\/code> ta \u0111\u01b0\u1ee3c <code>x = \u03c0\/4 + k\u03c0<\/code> (v\u1edbi k thu\u1ed9c Z).<br \/>\n\u2022 Do \u0111\u00f3 \u0111i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh l\u00e0 <code>x kh\u00e1c \u03c0\/4 + k\u03c0<\/code> (k thu\u1ed9c Z).<br \/>\n\u2022 V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 l\u00e0:<\/p>\n<p style=\"margin: 0;\"><code>D = R \\ {\u03c0\/4 + k\u03c0, k \u2208 Z}<\/code>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 6: T\u00cdNH TU\u1ea6N HO\u00c0N C\u1ee6A H\u00c0M S\u1ed0 L\u01af\u1ee2NG GI\u00c1C --><\/p>\n<h2 id=\"muc-8-tinh-tuan-hoan-cua-ham-so-luong-giac\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">T\u00ednh tu\u1ea7n ho\u00e0n c\u1ee7a h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">Kh\u00e1i ni\u1ec7m chu k\u00ec v\u00e0 c\u00e1ch x\u00e1c \u0111\u1ecbnh chu k\u00ec h\u00e0m s\u1ed1 h\u1ee3p<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>\u0110\u1ecbnh ngh\u0129a:<\/strong> H\u00e0m s\u1ed1 y = f(x) \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 h\u00e0m s\u1ed1 tu\u1ea7n ho\u00e0n n\u1ebfu t\u1ed3n t\u1ea1i s\u1ed1 T kh\u00e1c 0 sao cho v\u1edbi m\u1ecdi x thu\u1ed9c t\u1eadp x\u00e1c \u0111\u1ecbnh th\u00ec x + T v\u00e0 x &#8211; T thu\u1ed9c t\u1eadp x\u00e1c \u0111\u1ecbnh, \u0111\u1ed3ng th\u1eddi f(x + T) = f(x). S\u1ed1 d\u01b0\u01a1ng T nh\u1ecf nh\u1ea5t th\u1ecfa m\u00e3n t\u00ednh ch\u1ea5t \u0111\u00f3 g\u1ecdi l\u00e0 chu k\u00ec c\u1ee7a h\u00e0m s\u1ed1.<\/li>\n<li><strong>Chu k\u00ec c\u1ee7a c\u00e1c h\u00e0m c\u01a1 b\u1ea3n:<\/strong>\n<ul>\n<li>C\u00e1c h\u00e0m s\u1ed1 y = sin x v\u00e0 y = cos x tu\u1ea7n ho\u00e0n v\u1edbi chu k\u00ec <code>T = 2\u03c0<\/code>.<\/li>\n<li>C\u00e1c h\u00e0m s\u1ed1 y = tan x v\u00e0 y = cot x tu\u1ea7n ho\u00e0n v\u1edbi chu k\u00ec <code>T = \u03c0<\/code>.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Chu k\u00ec c\u1ee7a h\u00e0m s\u1ed1 d\u1ea1ng m\u1edf r\u1ed9ng:<\/strong>\n<ul>\n<li>H\u00e0m s\u1ed1 <code>y = sin(ax + b)<\/code> ho\u1eb7c <code>y = cos(ax + b)<\/code> c\u00f3 chu k\u00ec l\u00e0 <code>T = 2\u03c0 \/ |a|<\/code>.<\/li>\n<li>H\u00e0m s\u1ed1 <code>y = tan(ax + b)<\/code> ho\u1eb7c <code>y = cot(ax + b)<\/code> c\u00f3 chu k\u00ec l\u00e0 <code>T = \u03c0 \/ |a|<\/code> (v\u1edbi a kh\u00e1c 0).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!-- B\u1ea2NG T\u00d3M T\u1eaeT CHU K\u00cc TU\u1ea6N HO\u00c0N --><\/p>\n<div style=\"overflow-x: auto; margin: 15px 0;\">\n<table style=\"table-layout: auto; font-size: 14px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: left;\">\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; color: #0056b3;\">H\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">C\u00f4ng th\u1ee9c chu k\u00ec T<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">V\u00ed d\u1ee5 minh h\u1ecda c\u1ee5 th\u1ec3<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = sin(3x)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">T = 2\u03c0 \/ 3<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">H\u1ec7 s\u1ed1 a = 3 n\u00ean T = 2\u03c0 \/ 3<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = cos(x \/ 2)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">T = 4\u03c0<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">H\u1ec7 s\u1ed1 a = 1\/2 n\u00ean T = 2\u03c0 \/ (1\/2) = 4\u03c0<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">y = tan(2x + 1)<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">T = \u03c0 \/ 2<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">H\u1ec7 s\u1ed1 a = 2 n\u00ean T = \u03c0 \/ 2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p khi t\u00ecm chu k\u00ec:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Qu\u00ean l\u1ea5y gi\u00e1 tr\u1ecb tuy\u1ec7t \u0111\u1ed1i c\u1ee7a h\u1ec7 s\u1ed1 a (v\u00ed d\u1ee5 v\u1edbi h\u00e0m y = sin(-3x) l\u1ea1i t\u00ednh chu k\u00ec th\u00e0nh s\u1ed1 \u00e2m) ho\u1eb7c nh\u1ea7m l\u1eabn gi\u1eefa chu k\u00ec c\u1ee7a h\u00e0m sin\/cos (2\u03c0) v\u1edbi h\u00e0m tan\/cot (\u03c0).<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: T\u00ecm chu k\u00ec tu\u1ea7n ho\u00e0n c\u1ee7a h\u00e0m s\u1ed1 h\u1ee3p<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> T\u00ecm chu k\u00ec T c\u1ee7a h\u00e0m s\u1ed1 sau: <code>y = 3 cos(4x - \u03c0\/3) + 2 sin(2x)<\/code>.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 H\u00e0m s\u1ed1 g\u1ed3m t\u1ed5ng c\u1ee7a hai h\u00e0m th\u00e0nh ph\u1ea7n:<br \/>\n&#8211; X\u00e9t h\u00e0m s\u1ed1 th\u1ee9 nh\u1ea5t <code>y1 = 3 cos(4x - \u03c0\/3)<\/code> c\u00f3 h\u1ec7 s\u1ed1 a = 4, chu k\u00ec l\u00e0 <code>T1 = 2\u03c0 \/ 4 = \u03c0 \/ 2<\/code>.<br \/>\n&#8211; X\u00e9t h\u00e0m s\u1ed1 th\u1ee9 hai <code>y2 = 2 sin(2x)<\/code> c\u00f3 h\u1ec7 s\u1ed1 a = 2, chu k\u00ec l\u00e0 <code>T2 = 2\u03c0 \/ 2 = \u03c0<\/code>.<\/p>\n<p style=\"margin: 0;\">\u2022 Chu k\u00ec c\u1ee7a h\u00e0m s\u1ed1 t\u1ed5ng y ch\u00ednh l\u00e0 b\u1ed9i chung nh\u1ecf nh\u1ea5t c\u1ee7a hai chu k\u00ec T1 v\u00e0 T2.<br \/>\n&#8211; Ta c\u00f3 BCNN c\u1ee7a (\u03c0 \/ 2) v\u00e0 \u03c0 ch\u00ednh l\u00e0 <strong>\u03c0<\/strong>.<br \/>\n\u2022 V\u1eady h\u00e0m s\u1ed1 \u0111\u00e3 cho tu\u1ea7n ho\u00e0n v\u1edbi chu k\u00ec <strong>T = \u03c0<\/strong>.<\/p>\n<\/div>\n<\/div>\n<p><!-- K\u1ebeT TH\u00daC: PH\u1ea6N I (TI\u1ebeP THEO) - H\u00c0M S\u1ed0 L\u01af\u1ee2NG GI\u00c1C TO\u00c1N 11 --><\/p>\n<p><!-- B\u1eaeT \u0110\u1ea6U: PH\u1ea6N I (2 H2 CU\u1ed0I) - PH\u01af\u01a0NG TR\u00ccNH V\u00c0 GTLN GTNN L\u01af\u1ee2NG GI\u00c1C TO\u00c1N 11 --><\/p>\n<div class=\"naviedu-content-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<p><!-- CH\u1ee6 \u0110\u1ec0 7: PH\u01af\u01a0NG TR\u00ccNH L\u01af\u1ee2NG GI\u00c1C C\u01a0 B\u1ea2N --><\/p>\n<h2 id=\"muc-9-phuong-trinh-luong-giac-co-ban\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">C\u00f4ng th\u1ee9c nghi\u1ec7m c\u01a1 b\u1ea3n v\u00e0 c\u00e1c tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t<\/h3>\n<p>Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n l\u00e0 n\u1ec1n t\u1ea3ng c\u1ed1t l\u00f5i \u0111\u1ec3 gi\u1ea3i quy\u1ebft c\u00e1c b\u00e0i to\u00e1n ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c ph\u1ee9c t\u1ea1p. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 b\u1ea3ng t\u1ed5ng h\u1ee3p c\u00f4ng th\u1ee9c nghi\u1ec7m t\u1ed5ng qu\u00e1t:<\/p>\n<p><!-- B\u1ea2NG C\u00d4NG TH\u1ee8C NGHI\u1ec6M C\u01a0 B\u1ea2N --><\/p>\n<div style=\"overflow-x: auto; margin: 15px 0;\">\n<table style=\"table-layout: auto; font-size: 14px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: left;\">\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; color: #0056b3;\">Ph\u01b0\u01a1ng tr\u00ecnh<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">\u0110i\u1ec1u ki\u1ec7n c\u00f3 nghi\u1ec7m<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">C\u00f4ng th\u1ee9c nghi\u1ec7m t\u1ed5ng qu\u00e1t (k \u2208 Z)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">sin x = m<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">|m| \u2264 1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">x = \u03b1 + k2\u03c0 ho\u1eb7c x = \u03c0 &#8211; \u03b1 + k2\u03c0 (v\u1edbi sin \u03b1 = m)<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">cos x = m<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">|m| \u2264 1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">x = \u00b1\u03b1 + k2\u03c0 (v\u1edbi cos \u03b1 = m)<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">tan x = m<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">V\u1edbi m\u1ecdi m \u2208 R<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">x = \u03b1 + k\u03c0 (v\u1edbi tan \u03b1 = m)<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">cot x = m<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">V\u1edbi m\u1ecdi m \u2208 R<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">x = \u03b1 + k\u03c0 (v\u1edbi cot \u03b1 = m)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><strong>C\u00e1c tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t c\u1ea7n ghi nh\u1edb:<\/strong><\/p>\n<ul style=\"padding-left: 20px;\">\n<li>sin x = 0 t\u01b0\u01a1ng \u0111\u01b0\u01a1ng x = k\u03c0 ; sin x = 1 t\u01b0\u01a1ng \u0111\u01b0\u01a1ng x = \u03c0\/2 + k2\u03c0 ; sin x = -1 t\u01b0\u01a1ng \u0111\u01b0\u01a1ng x = -\u03c0\/2 + k2\u03c0 (k \u2208 Z).<\/li>\n<li>cos x = 0 t\u01b0\u01a1ng \u0111\u01b0\u01a1ng x = \u03c0\/2 + k\u03c0 ; cos x = 1 t\u01b0\u01a1ng \u0111\u01b0\u01a1ng x = k2\u03c0 ; cos x = -1 t\u01b0\u01a1ng \u0111\u01b0\u01a1ng x = \u03c0 + k2\u03c0 (k \u2208 Z).<\/li>\n<li>Ph\u01b0\u01a1ng tr\u00ecnh t\u01b0\u01a1ng \u0111\u01b0\u01a1ng: sin u = sin v t\u01b0\u01a1ng \u0111\u01b0\u01a1ng u = v + k2\u03c0 ho\u1eb7c u = \u03c0 &#8211; v + k2\u03c0 ; cos u = cos v t\u01b0\u01a1ng \u0111\u01b0\u01a1ng u = \u00b1v + k2\u03c0 (k \u2208 Z).<\/li>\n<\/ul>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p khi gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Qu\u00ean \u0111i\u1ec1u ki\u1ec7n c\u00f3 nghi\u1ec7m c\u1ee7a sin v\u00e0 cos (|m| \u2264 1), d\u1eabn \u0111\u1ebfn vi\u1ec7c gi\u1ea3i ra g\u00f3c \u03b1 cho c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh v\u00f4 nghi\u1ec7m (nh\u01b0 cos x = 2); ho\u1eb7c qu\u00ean ghi k\u00e8m \u0111u\u00f4i &#8220;k thu\u1ed9c Z&#8221; trong h\u1ecd nghi\u1ec7m.<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU 1 --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu 1 (T\u1ef1 lu\u1eadn): Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh sau tr\u00ean t\u1eadp s\u1ed1 th\u1ef1c: <code>2 cos(2x - \u03c0\/3) - 1 = 0<\/code>.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 Bi\u1ebfn \u0111\u1ed5i ph\u01b0\u01a1ng tr\u00ecnh v\u1ec1 d\u1ea1ng c\u01a1 b\u1ea3n:<br \/>\n<code>2 cos(2x - \u03c0\/3) = 1<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>cos(2x - \u03c0\/3) = 1\/2<\/code>.<br \/>\n\u2022 V\u00ec 1\/2 ch\u00ednh l\u00e0 cos(\u03c0\/3), ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh:<br \/>\n<code>cos(2x - \u03c0\/3) = cos(\u03c0\/3)<\/code>.<br \/>\n\u2022 \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh c\u00f4sin:<br \/>\n<code>2x - \u03c0\/3 = \u00b1(\u03c0\/3) + k2\u03c0<\/code> (v\u1edbi k thu\u1ed9c Z).<br \/>\n\u2022 Ta chia th\u00e0nh hai tr\u01b0\u1eddng h\u1ee3p:<br \/>\n&#8211; Tr\u01b0\u1eddng h\u1ee3p 1:<\/p>\n<p style=\"margin: 0;\"><code>2x - \u03c0\/3 = \u03c0\/3 + k2\u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>2x = 2\u03c0\/3 + k2\u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>x = \u03c0\/3 + k\u03c0<\/code>.<br \/>\n&#8211; Tr\u01b0\u1eddng h\u1ee3p 2: <code>2x - \u03c0\/3 = -\u03c0\/3 + k2\u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>2x = k2\u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>x = k\u03c0<\/code>.<br \/>\n\u2022 <em>\u0110\u00e1p s\u1ed1:<\/em> <strong>x = \u03c0\/3 + k\u03c0<\/strong> v\u00e0 <strong>x = k\u03c0<\/strong> (v\u1edbi k \u2208 Z).<\/p>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU 2 (D\u1ea0NG TR\u1ea2 L\u1edcI NG\u1eaeN) --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu 2 (Tr\u1ea3 l\u1eddi ng\u1eafn): \u0110\u1ebfm s\u1ed1 nghi\u1ec7m tr\u00ean kho\u1ea3ng cho tr\u01b0\u1edbc<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> T\u00ecm s\u1ed1 nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh <code>sin(3x) = 1\/2<\/code> thu\u1ed9c kho\u1ea3ng t\u1eeb 0 \u0111\u1ebfn \u03c0.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh <code>sin(3x) = 1\/2<\/code>:<br \/>\nV\u00ec 1\/2 = sin(\u03c0\/6), ta c\u00f3 hai h\u1ecd nghi\u1ec7m:<br \/>\n&#8211; <code>3x = \u03c0\/6 + k2\u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>x = \u03c0\/18 + (k2\u03c0)\/3<\/code>.<br \/>\n&#8211; <code>3x = \u03c0 - \u03c0\/6 + k2\u03c0 = 5\u03c0\/6 + k2\u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>x = 5\u03c0\/18 + (k2\u03c0)\/3<\/code> (v\u1edbi k thu\u1ed9c Z).<br \/>\n\u2022 X\u00e9t \u0111i\u1ec1u ki\u1ec7n nghi\u1ec7m thu\u1ed9c kho\u1ea3ng <code>(0; \u03c0)<\/code>:<br \/>\n&#8211; V\u1edbi h\u1ecd nghi\u1ec7m th\u1ee9 nh\u1ea5t: <code>0 &lt; \u03c0\/18 + k2\u03c0\/3 &lt; \u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>-1\/12 &lt; k &lt; 17\/12<\/code>.<\/p>\n<p style=\"margin: 0;\">Do k thu\u1ed9c Z n\u00ean <code>k nh\u1eadn c\u00e1c gi\u00e1 tr\u1ecb: 0, 1<\/code> (cho 2 nghi\u1ec7m).<br \/>\n&#8211; V\u1edbi h\u1ecd nghi\u1ec7m th\u1ee9 hai: <code>0 &lt; 5\u03c0\/18 + k2\u03c0\/3 &lt; \u03c0<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>-5\/12 &lt; k &lt; 13\/12<\/code>. Do k thu\u1ed9c Z n\u00ean <code>k nh\u1eadn c\u00e1c gi\u00e1 tr\u1ecb: 0, 1<\/code> (cho 2 nghi\u1ec7m).<br \/>\n\u2022 T\u1ed5ng s\u1ed1 nghi\u1ec7m th\u1ecfa m\u00e3n tr\u00ean kho\u1ea3ng t\u1eeb 0 \u0111\u1ebfn \u03c0 l\u00e0 2 + 2 = 4 nghi\u1ec7m.<br \/>\n\u2022 <em>\u0110\u00e1p s\u1ed1:<\/em> <strong>4 nghi\u1ec7m<\/strong>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 8: GI\u00c1 TR\u1eca L\u1edaN NH\u1ea4T, NH\u1ece NH\u1ea4T C\u1ee6A BI\u1ec2U TH\u1ee8C L\u01af\u1ee2NG GI\u00c1C --><\/p>\n<h2 id=\"muc-10-gia-tri-lon-nhat-nho-nhat-cua-bieu-thuc-luong-giac\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t, nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">C\u00e1c d\u1ea1ng to\u00e1n t\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t v\u00e0 nh\u1ecf nh\u1ea5t tr\u1ecdng t\u00e2m<\/h3>\n<p>B\u00e0i to\u00e1n t\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t (GTLN) v\u00e0 gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t (GTNN) c\u1ee7a h\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c th\u01b0\u1eddng xu\u1ea5t hi\u1ec7n trong ph\u1ea7n tr\u1eafc nghi\u1ec7m v\u00e0 t\u1ef1 lu\u1eadn v\u1edbi ba d\u1ea1ng ch\u00ednh:<\/p>\n<p><!-- B\u1ea2NG T\u00d3M T\u1eaeT C\u00c1C D\u1ea0NG GTLN - GTNN --><\/p>\n<div style=\"overflow-x: auto; margin: 15px 0;\">\n<table style=\"table-layout: auto; font-size: 14px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: left;\">\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; color: #0056b3;\">D\u1ea1ng to\u00e1n bi\u1ec3u th\u1ee9c<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">Ph\u01b0\u01a1ng ph\u00e1p \/ \u0110\u00e1nh gi\u00e1 c\u1ed1t l\u00f5i<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">C\u00f4ng th\u1ee9c bi\u00ean \u0111\u1ed9 GTLN &#8211; GTNN<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">D\u1ea1ng c\u01a1 b\u1ea3n: y = a sin x + b<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">D\u1ef1a v\u00e0o t\u00ednh ch\u1ea5t -1 \u2264 sin x \u2264 1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">Max = |a| + b; Min = \u2212|a| + b<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">D\u1ea1ng b\u1ed5 tr\u1ee3: y = a sin x + b cos x<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">\u00c1p d\u1ee5ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c Bunhiac\u1ed1pxki<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">-\u221a(a<sup>2<\/sup> + b<sup>2<\/sup>) \u2264 y \u2264 \u221a(a<sup>2<\/sup> + b<sup>2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0; font-weight: bold;\">D\u1ea1ng b\u1eadc hai theo sin ho\u1eb7c cos<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">\u0110\u1eb7t \u1ea9n ph\u1ee5 t = sin x (ho\u1eb7c cos x), x\u00e9t t thu\u1ed9c [-1; 1]<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 10px; border: 1px solid #cbd5e0;\">Kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1 b\u1eadc hai tr\u00ean \u0111o\u1ea1n [-1; 1]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng \u0433\u0440\u0430\u0444 khi t\u00ecm GTLN &#8211; GTNN:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Qu\u00ean ki\u1ec3m tra \u0111i\u1ec1u ki\u1ec7n \u0111\u1ea1t \u0111\u01b0\u1ee3c d\u1ea5u b\u1eb1ng (\u0111\u1ea3m b\u1ea3o t\u1ed3n t\u1ea1i gi\u00e1 tr\u1ecb c\u1ee7a x \u0111\u1ec3 h\u00e0m s\u1ed1 \u0111\u1ea1t \u0111\u01b0\u1ee3c GTLN ho\u1eb7c GTNN \u0111\u00f3) ho\u1eb7c qu\u00ean m\u1ea5t mi\u1ec1n gi\u00e1 tr\u1ecb c\u1ee7a \u1ea9n ph\u1ee5 t khi \u0111\u1eb7t t = sin x (t ch\u1ec9 thu\u1ed9c \u0111o\u1ea1n t\u1eeb -1 \u0111\u1ebfn 1 ch\u1ee9 kh\u00f4ng ph\u1ea3i to\u00e0n b\u1ed9 t\u1eadp R).<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU 1 --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu 1 (T\u1ef1 lu\u1eadn): T\u00ecm GTLN GTNN d\u1ea1ng t\u1ed5ng sin v\u00e0 cos<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> T\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t v\u00e0 gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a h\u00e0m s\u1ed1: <code>y = 3 sin x - 4 cos x + 5<\/code>.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 X\u00e9t bi\u1ec3u th\u1ee9c ch\u1ee9a sin v\u00e0 cos: <code>3 sin x - 4 cos x<\/code>.<br \/>\n\u2022 \u00c1p d\u1ee5ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c bi\u00ean \u0111\u1ed9 \u0111\u1ed1i v\u1edbi h\u00e0m s\u1ed1 d\u1ea1ng a sin x + b cos x v\u1edbi a = 3, b = -4:<br \/>\n&#8211; Ta c\u00f3: <code>-\u221a(3<sup>2<\/sup> + (-4)<sup>2<\/sup>) \u2264 3 sin x - 4 cos x \u2264 \u221a(3<sup>2<\/sup> + (-4)<sup>2<\/sup>)<\/code><br \/>\n&#8211; T\u01b0\u01a1ng \u0111\u01b0\u01a1ng:<\/p>\n<p style=\"margin: 0;\"><code>-\u221a25 \u2264 3 sin x - 4 cos x \u2264 \u221a25<\/code> hay <code>-5 \u2264 3 sin x - 4 cos x \u2264 5<\/code>.<br \/>\n\u2022 C\u1ed9ng th\u00eam 5 v\u00e0o t\u1ea5t c\u1ea3 c\u00e1c v\u1ebf c\u1ee7a b\u1ea5t \u0111\u1eb3ng th\u1ee9c \u0111\u1ec3 \u0111\u01b0\u1ee3c h\u00e0m s\u1ed1 y:<\/p>\n<p style=\"margin: 0;\"><code>-5 + 5 \u2264 3 sin x - 4 cos x + 5 \u2264 5 + 5<\/code> t\u01b0\u01a1ng \u0111\u01b0\u01a1ng <code>0 \u2264 y \u2264 10<\/code>.<br \/>\n\u2022 T\u1eeb \u0111\u00e2y ta suy ra gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a y b\u1eb1ng 0 (khi d\u1ea5u b\u1eb1ng x\u1ea3y ra) v\u00e0 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a y b\u1eb1ng 10.<br \/>\n\u2022 <em>\u0110\u00e1p s\u1ed1:<\/em> <strong>GTNN = 0<\/strong> v\u00e0 <strong>GTLN = 10<\/strong>.<\/p>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU 2 --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu 2 (T\u1ef1 lu\u1eadn): T\u00ecm GTLN GTNN b\u1eb1ng ph\u01b0\u01a1ng ph\u00e1p \u1ea9n ph\u1ee5 b\u1eadc hai<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> T\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t v\u00e0 gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a h\u00e0m s\u1ed1: <code>y = sin<sup>2<\/sup>x - 4 sin x + 3<\/code>.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 \u0110\u1eb7t \u1ea9n ph\u1ee5 <code>t = sin x<\/code>. Do sin x dao \u0111\u1ed9ng tr\u00ean t\u1eadp s\u1ed1 th\u1ef1c n\u00ean \u0111i\u1ec1u ki\u1ec7n c\u1ee7a t l\u00e0: <code>t thu\u1ed9c [-1; 1]<\/code>.<br \/>\n\u2022 Khi \u0111\u00f3 h\u00e0m s\u1ed1 tr\u1edf th\u00e0nh h\u00e0m s\u1ed1 b\u1eadc hai theo bi\u1ebfn t:<br \/>\n<code>y = f(t) = t<sup>2<\/sup> - 4t + 3<\/code> tr\u00ean \u0111o\u1ea1n <code>[-1; 1]<\/code>.<br \/>\n\u2022 \u0110\u1ea1o h\u00e0m ho\u1eb7c x\u00e9t \u0111\u1ec9nh c\u1ee7a parabol h\u00e0m s\u1ed1 f(t):<br \/>\n&#8211; Ho\u00e0nh \u0111\u1ed9 \u0111\u1ec9nh c\u1ee7a parabol l\u00e0 t = -(-4) \/ (2 \u00b7 1) = 2.<\/p>\n<p style=\"margin: 0;\">Gi\u00e1 tr\u1ecb n\u00e0y kh\u00f4ng thu\u1ed9c \u0111o\u1ea1n x\u00e9t [-1; 1].<br \/>\n&#8211; Do h\u1ec7 s\u1ed1 c\u1ee7a t<sup>2<\/sup> d\u01b0\u01a1ng (h\u01b0\u1edbng b\u1ec1 l\u00f5m quay l\u00ean), h\u00e0m s\u1ed1 ngh\u1ecbch bi\u1ebfn tr\u00ean \u0111o\u1ea1n [-1; 1].<br \/>\n\u2022 Ta t\u00ednh gi\u00e1 tr\u1ecb t\u1ea1i hai \u0111\u1ea7u m\u00fat c\u1ee7a \u0111o\u1ea1n [-1; 1]:<br \/>\n&#8211; T\u1ea1i t = -1: <code>f(-1) = (-1)<sup>2<\/sup> - 4(-1) + 3 = 1 + 4 + 3 = 8<\/code>.<br \/>\n&#8211; T\u1ea1i t = 1:<\/p>\n<p style=\"margin: 0;\"><code>f(1) = 1<sup>2<\/sup> - 4(1) + 3 = 1 - 4 + 3 = 0<\/code>.<br \/>\n\u2022 So s\u00e1nh c\u00e1c gi\u00e1 tr\u1ecb, ta c\u00f3 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a y b\u1eb1ng 8 (khi sin x = -1) v\u00e0 gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a y b\u1eb1ng 0 (khi sin x = 1).<br \/>\n\u2022 <em>\u0110\u00e1p s\u1ed1:<\/em> <strong>GTNN = 0<\/strong> v\u00e0 <strong>GTLN = 8<\/strong>.<\/p>\n<\/div>\n<\/div>\n<p><!-- K\u1ebeT TH\u00daC: PH\u1ea6N I (2 H2 CU\u1ed0I) - PH\u01af\u01a0NG TR\u00ccNH V\u00c0 GTLN GTNN L\u01af\u1ee2NG GI\u00c1C TO\u00c1N 11 --><\/p>\n<p><!-- B\u1eaeT \u0110\u1ea6U: PH\u1ea6N II - \u0110\u01af\u1edcNG TH\u1eb2NG V\u00c0 M\u1eb6T PH\u1eb2NG TRONG KH\u00d4NG GIAN TO\u00c1N 11 --><\/p>\n<div class=\"naviedu-content-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<p><!-- BANNER PH\u1ea6N II --><\/p>\n<div style=\"background: linear-gradient(135deg, #0056b3, #0080ff); color: #ffffff; padding: 18px 24px; border-radius: 8px; margin-top: 35px; margin-bottom: 30px;\">\n<h2 id=\"muc-11-phan-ii-duong-thang-va-mat-phang-trong-khong-gian\" style=\"color: #ffffff; margin: 0; font-size: 22px; text-transform: uppercase; letter-spacing: 0.5px;\">Ph\u1ea7n II: \u0110\u01b0\u1eddng Th\u1eb3ng v\u00e0 M\u1eb7t Ph\u1eb3ng Trong Kh\u00f4ng Gian<\/h2>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 1: \u0110\u01af\u1edcNG TH\u1eb2NG V\u00c0 M\u1eb6T PH\u1eb2NG TRONG KH\u00d4NG GIAN --><\/p>\n<p style=\"margin: 0 0 16px 0;\">H\u00ecnh h\u1ecdc kh\u00f4ng gian l\u00e0 ph\u1ea7n m\u1edbi ho\u00e0n to\u00e0n trong \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11. H\u00e3y t\u1eadp v\u1ebd h\u00ecnh \u0111\u00fang quy t\u1eafc tr\u01b0\u1edbc, r\u1ed3i m\u1edbi luy\u1ec7n t\u00ecm giao tuy\u1ebfn v\u00e0 giao \u0111i\u1ec3m.<\/p>\n<h2 id=\"muc-12-duong-thang-va-mat-phang-trong-khong-gian\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">\u0110\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng trong kh\u00f4ng gian<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">C\u00e1c t\u00ednh ch\u1ea5t th\u1eeba nh\u1eadn v\u00e0 c\u00e1ch x\u00e1c \u0111\u1ecbnh m\u1eb7t ph\u1eb3ng<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>C\u00e1c kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n:<\/strong> \u0110i\u1ec3m, \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng l\u00e0 c\u00e1c kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n kh\u00f4ng \u0111\u1ecbnh ngh\u0129a trong h\u00ecnh h\u1ecdc kh\u00f4ng gian.<\/li>\n<li><strong>B\u1ed1n c\u00e1ch x\u00e1c \u0111\u1ecbnh m\u1ed9t m\u1eb7t ph\u1eb3ng duy nh\u1ea5t:<\/strong>\n<ol>\n<li>Qua ba \u0111i\u1ec3m kh\u00f4ng th\u1eb3ng h\u00e0ng.<\/li>\n<li>Qua m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1ed9t \u0111i\u1ec3m kh\u00f4ng thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng \u0111\u00f3.<\/li>\n<li>Qua hai \u0111\u01b0\u1eddng th\u1eb3ng c\u1eaft nhau.<\/li>\n<li>Qua hai \u0111\u01b0\u1eddng th\u1eb3ng song song.<\/li>\n<\/ol>\n<\/li>\n<li><strong>H\u00ecnh ch\u00f3p v\u00e0 t\u1ee9 di\u1ec7n:<\/strong> H\u00ecnh ch\u00f3p c\u00f3 \u0111\u00e1y l\u00e0 \u0111a gi\u00e1c v\u00e0 c\u00e1c m\u1eb7t b\u00ean l\u00e0 c\u00e1c tam gi\u00e1c c\u00f3 chung \u0111\u1ec9nh. T\u1ee9 di\u1ec7n l\u00e0 h\u00ecnh ch\u00f3p c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c (g\u1ed3m 4 m\u1eb7t \u0111\u1ec1u l\u00e0 tam gi\u00e1c).<\/li>\n<li><strong>Quy t\u1eafc v\u1ebd h\u00ecnh kh\u00f4ng gian:<\/strong> Khi v\u1ebd h\u00ecnh bi\u1ec3u di\u1ec5n c\u1ee7a c\u00e1c h\u00ecnh kh\u1ed1i trong kh\u00f4ng gian, c\u00e1c \u0111\u01b0\u1eddng nh\u00ecn th\u1ea5y v\u1ebd n\u00e9t li\u1ec1n, c\u00f2n c\u00e1c <strong>\u0111\u01b0\u1eddng b\u1ecb che khu\u1ea5t ph\u1ea3i v\u1ebd b\u1eb1ng n\u00e9t \u0111\u1ee9t<\/strong>.<\/li>\n<\/ul>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p khi v\u1ebd h\u00ecnh kh\u00f4ng gian:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>V\u1ebd n\u00e9t li\u1ec1n cho c\u00e1c c\u1ea1nh khu\u1ea5t \u1edf ph\u00eda sau h\u00ecnh ch\u00f3p ho\u1eb7c nh\u1ea7m l\u1eabn ba \u0111i\u1ec3m th\u1eb3ng h\u00e0ng l\u1ea1i c\u00f3 th\u1ec3 x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c m\u1ed9t m\u1eb7t ph\u1eb3ng ri\u00eang bi\u1ec7t.<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU (D\u1ea0NG \u0110\u00daNG\/SAI A, B, C, D) --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: Nh\u1eadn di\u1ec7n t\u00ednh ch\u1ea5t h\u00ecnh h\u1ecdc kh\u00f4ng gian<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> Cho h\u00ecnh ch\u00f3p S.ABCD. X\u00e9t t\u00ednh \u0111\u00fang sai c\u1ee7a c\u00e1c ph\u00e1t bi\u1ec3u sau:<br \/>\na) Ba \u0111i\u1ec3m b\u1ea5t k\u00ec trong kh\u00f4ng gian lu\u00f4n x\u00e1c \u0111\u1ecbnh m\u1ed9t m\u1eb7t ph\u1eb3ng.<br \/>\nb) Hai \u0111\u01b0\u1eddng th\u1eb3ng c\u1eaft nhau x\u00e1c \u0111\u1ecbnh duy nh\u1ea5t m\u1ed9t m\u1eb7t ph\u1eb3ng.<br \/>\nc) C\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng b\u1ecb che khu\u1ea5t trong h\u00ecnh v\u1ebd kh\u00f4ng gian \u0111\u01b0\u1ee3c ph\u00e9p v\u1ebd n\u00e9t li\u1ec1n.<br \/>\nd) H\u00ecnh t\u1ee9 di\u1ec7n l\u00e0 h\u00ecnh ch\u00f3p c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 <strong>\u00dd a:<\/strong> Ba \u0111i\u1ec3m th\u1eb3ng h\u00e0ng ch\u1ec9 x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c duy nh\u1ea5t m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng ch\u1ee9 kh\u00f4ng x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c m\u1eb7t ph\u1eb3ng. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>sai<\/strong>.<br \/>\n\u2022 <strong>\u00dd b:<\/strong> Theo t\u00ednh ch\u1ea5t c\u01a1 b\u1ea3n, hai \u0111\u01b0\u1eddng th\u1eb3ng c\u1eaft nhau x\u00e1c \u0111\u1ecbnh duy nh\u1ea5t m\u1ed9t m\u1eb7t ph\u1eb3ng.<\/p>\n<p style=\"margin: 0;\">Ph\u00e1t bi\u1ec3u n\u00e0y <strong>\u0111\u00fang<\/strong>.<br \/>\n\u2022 <strong>\u00dd c:<\/strong> Quy t\u1eafc v\u1ebd h\u00ecnh b\u1eaft bu\u1ed9c c\u00e1c \u0111\u01b0\u1eddng khu\u1ea5t ph\u1ea3i v\u1ebd n\u00e9t \u0111\u1ee9t. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>sai<\/strong>.<br \/>\n\u2022 <strong>\u00dd d:<\/strong> T\u1ee9 di\u1ec7n ch\u00ednh l\u00e0 h\u00ecnh ch\u00f3p tam gi\u00e1c v\u1edbi 4 m\u1eb7t \u0111\u1ec1u l\u00e0 tam gi\u00e1c. Ph\u00e1t bi\u1ec3u n\u00e0y <strong>\u0111\u00fang<\/strong>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 2: T\u00ccM GIAO TUY\u1ebeN C\u1ee6A HAI M\u1eb6T PH\u1eb2NG --><\/p>\n<h2 id=\"muc-13-tim-giao-tuyen-cua-hai-mat-phang\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">T\u00ecm giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">Ph\u01b0\u01a1ng ph\u00e1p t\u00ecm giao tuy\u1ebfn<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>Ph\u01b0\u01a1ng ph\u00e1p 1 (T\u00ecm 2 \u0111i\u1ec3m chung):<\/strong> T\u00ecm 2 \u0111i\u1ec3m ph\u00e2n bi\u1ec7t c\u00f9ng thu\u1ed9c \u0111\u1ed3ng th\u1eddi c\u1ea3 hai m\u1eb7t ph\u1eb3ng. \u0110\u01b0\u1eddng th\u1eb3ng n\u1ed1i hai \u0111i\u1ec3m chung \u0111\u00f3 ch\u00ednh l\u00e0 giao tuy\u1ebfn.<\/li>\n<li><strong>Ph\u01b0\u01a1ng ph\u00e1p 2 (T\u00ecm 1 \u0111i\u1ec3m chung + ph\u01b0\u01a1ng song song):<\/strong> T\u00ecm 1 \u0111i\u1ec3m chung c\u1ee7a hai m\u1eb7t ph\u1eb3ng, k\u1ebft h\u1ee3p v\u1edbi t\u00ednh ch\u1ea5t ch\u1ee9a hai \u0111\u01b0\u1eddng th\u1eb3ng song song th\u00ec giao tuy\u1ebfn \u0111i qua \u0111i\u1ec3m chung v\u00e0 song song v\u1edbi hai \u0111\u01b0\u1eddng th\u1eb3ng \u0111\u00f3.<\/li>\n<\/ul>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p khi t\u00ecm giao tuy\u1ebfn:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Ch\u1ecdn nh\u1ea7m c\u00e1c \u0111i\u1ec3m kh\u00f4ng \u0111\u1ed3ng ph\u1eb3ng ho\u1eb7c kh\u00f4ng ch\u1ee9ng minh \u0111i\u1ec3m \u0111\u00f3 th\u1ef1c s\u1ef1 thu\u1ed9c c\u1ea3 hai m\u1eb7t ph\u1eb3ng tr\u01b0\u1edbc khi n\u1ed1i th\u00e0nh giao tuy\u1ebfn.<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: T\u00ecm giao tuy\u1ebfn c\u1ee7a h\u00ecnh ch\u00f3p \u0111\u00e1y h\u00ecnh b\u00ecnh h\u00e0nh<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y ABCD l\u00e0 h\u00ecnh b\u00ecnh h\u00e0nh t\u00e2m O. T\u00ecm giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng (SAC) v\u00e0 (SBD).<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 B\u01b0\u1edbc 1: T\u00ecm \u0111i\u1ec3m chung th\u1ee9 nh\u1ea5t:<br \/>\n&#8211; \u0110\u1ec9nh S r\u00f5 r\u00e0ng thu\u1ed9c m\u1eb7t ph\u1eb3ng (SAC) v\u00e0 c\u0169ng thu\u1ed9c m\u1eb7t ph\u1eb3ng (SBD). Suy ra <code>S l\u00e0 \u0111i\u1ec3m chung th\u1ee9 nh\u1ea5t<\/code>.<br \/>\n\u2022 B\u01b0\u1edbc 2:<\/p>\n<p style=\"margin: 0;\">T\u00ecm \u0111i\u1ec3m chung th\u1ee9 hai:<br \/>\n&#8211; Trong m\u1eb7t ph\u1eb3ng \u0111\u00e1y (ABCD), g\u1ecdi O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a hai \u0111\u01b0\u1eddng ch\u00e9o AC v\u00e0 BD.<br \/>\n&#8211; V\u00ec O thu\u1ed9c AC m\u00e0 AC n\u1eb1m trong (SAC) n\u00ean <code>O thu\u1ed9c (SAC)<\/code>.<br \/>\n&#8211; V\u00ec O thu\u1ed9c BD m\u00e0 BD n\u1eb1m trong (SBD) n\u00ean <code>O thu\u1ed9c (SBD)<\/code>.<br \/>\n&#8211; Do \u0111\u00f3 <code>O l\u00e0 \u0111i\u1ec3m chung th\u1ee9 hai<\/code> c\u1ee7a hai m\u1eb7t ph\u1eb3ng.<br \/>\n\u2022 B\u01b0\u1edbc 3: K\u1ebft lu\u1eadn giao tuy\u1ebfn:<br \/>\n&#8211; \u0110\u01b0\u1eddng th\u1eb3ng n\u1ed1i hai \u0111i\u1ec3m chung S v\u00e0 O ch\u00ednh l\u00e0 giao tuy\u1ebfn. V\u1eady <code>(SAC) giao (SBD) = SO<\/code>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 3: T\u00ccM GIAO \u0110I\u1ec2M C\u1ee6A \u0110\u01af\u1edcNG TH\u1eb2NG V\u00c0 M\u1eb6T PH\u1eb2NG --><\/p>\n<h2 id=\"muc-14-tim-giao-diem-cua-duong-thang-va-mat-phang\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">T\u00ecm giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">Ph\u01b0\u01a1ng ph\u00e1p gi\u1ea3i to\u00e1n<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>Quy t\u1eafc t\u00ecm giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d v\u00e0 m\u1eb7t ph\u1eb3ng (P):<\/strong>\n<ol>\n<li>Ch\u1ecdn m\u1ed9t m\u1eb7t ph\u1eb3ng ph\u1ee5 (Q) ch\u1ee9a \u0111\u01b0\u1eddng th\u1eb3ng d sao cho (Q) d\u1ec5 d\u00e0ng t\u00ecm \u0111\u01b0\u1ee3c giao tuy\u1ebfn v\u1edbi m\u1eb7t ph\u1eb3ng (P).<\/li>\n<li>T\u00ecm giao tuy\u1ebfn d&#8217; c\u1ee7a hai m\u1eb7t ph\u1eb3ng (P) v\u00e0 (Q).<\/li>\n<li>Giao \u0111i\u1ec3m c\u1ea7n t\u00ecm ch\u00ednh l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng giao tuy\u1ebfn d&#8217; trong m\u1eb7t ph\u1eb3ng (Q).<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p khi t\u00ecm giao \u0111i\u1ec3m:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Ch\u1ecdn m\u1eb7t ph\u1eb3ng ph\u1ee5 kh\u00f4ng ch\u1ee9a \u0111\u01b0\u1eddng th\u1eb3ng d ho\u1eb7c ch\u1ecdn m\u1eb7t ph\u1eb3ng ph\u1ee5 m\u00e0 giao tuy\u1ebfn v\u1edbi m\u1eb7t ph\u1eb3ng (P) kh\u00f3 x\u00e1c \u0111\u1ecbnh ho\u1eb7c song song v\u1edbi d.<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: T\u00ecm giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y ABCD l\u00e0 h\u00ecnh b\u00ecnh h\u00e0nh. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u1ea1nh SC. T\u00ecm giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng AM v\u1edbi m\u1eb7t ph\u1eb3ng (SBD).<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 B\u01b0\u1edbc 1: Ch\u1ecdn m\u1eb7t ph\u1eb3ng ph\u1ee5 ch\u1ee9a \u0111\u01b0\u1eddng th\u1eb3ng AM.<br \/>\n&#8211; Ta ch\u1ecdn m\u1eb7t ph\u1eb3ng ph\u1ee5 ch\u1ee9a AM ch\u00ednh l\u00e0 m\u1eb7t ph\u1eb3ng (SAC).<br \/>\n\u2022 B\u01b0\u1edbc 2: T\u00ecm giao tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng ph\u1ee5 (SAC) v\u00e0 m\u1eb7t ph\u1eb3ng (SBD).<br \/>\n&#8211; Nh\u01b0 v\u00ed d\u1ee5 tr\u01b0\u1edbc, ta \u0111\u00e3 c\u00f3 giao tuy\u1ebfn c\u1ee7a (SAC) v\u00e0 (SBD) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng SO (v\u1edbi O l\u00e0 t\u00e2m h\u00ecnh b\u00ecnh h\u00e0nh ABCD).<br \/>\n\u2022 B\u01b0\u1edbc 3:<\/p>\n<p style=\"margin: 0;\">X\u00e1c \u0111\u1ecbnh giao \u0111i\u1ec3m trong m\u1eb7t ph\u1eb3ng ph\u1ee5 (SAC):<br \/>\n&#8211; Trong m\u1eb7t ph\u1eb3ng (SAC), g\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng AM v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng SO.<br \/>\n&#8211; V\u00ec I thu\u1ed9c SO m\u00e0 SO n\u1eb1m trong (SBD) n\u00ean <code>I thu\u1ed9c (SBD)<\/code>.<br \/>\n&#8211; V\u00ec I thu\u1ed9c AM n\u00ean giao \u0111i\u1ec3m c\u1ee7a AM v\u00e0 (SBD) ch\u00ednh l\u00e0 \u0111i\u1ec3m I.<br \/>\n\u2022 <em>\u0110\u00e1p s\u1ed1:<\/em> Giao \u0111i\u1ec3m l\u00e0 <strong>\u0111i\u1ec3m I (giao c\u1ee7a AM v\u00e0 SO)<\/strong>.<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 4: PH\u00c9P CHI\u1ebeU SONG SONG --><\/p>\n<h2 id=\"muc-15-phep-chieu-song-song\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">Ph\u00e9p chi\u1ebfu song song<\/h2>\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 20px;\">Kh\u00e1i ni\u1ec7m, t\u00ednh ch\u1ea5t v\u00e0 h\u00ecnh bi\u1ec3u di\u1ec5n<\/h3>\n<ul style=\"padding-left: 20px;\">\n<li><strong>Kh\u00e1i ni\u1ec7m:<\/strong> Cho m\u1eb7t ph\u1eb3ng (P) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng l c\u1eaft (P). Ph\u00e9p chi\u1ebfu song song theo ph\u01b0\u01a1ng l bi\u1ebfn m\u1ed7i \u0111i\u1ec3m M trong kh\u00f4ng gian th\u00e0nh \u0111i\u1ec3m M&#8217; tr\u00ean m\u1eb7t ph\u1eb3ng (P) sao cho \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua M song song ho\u1eb7c tr\u00f9ng v\u1edbi l c\u1eaft (P) t\u1ea1i M&#8217;.<\/li>\n<li><strong>C\u00e1c t\u00ednh ch\u1ea5t quan tr\u1ecdng c\u1ee7a ph\u00e9p chi\u1ebfu song song:<\/strong>\n<ul>\n<li>B\u1ea3o to\u00e0n t\u00ednh th\u1eb3ng h\u00e0ng (ba \u0111i\u1ec3m th\u1eb3ng h\u00e0ng chi\u1ebfu th\u00e0nh ba \u0111i\u1ec3m th\u1eb3ng h\u00e0ng).<\/li>\n<li>B\u1ea3o to\u00e0n t\u00ednh song song ho\u1eb7c c\u1eaft nhau c\u1ee7a hai \u0111\u01b0\u1eddng th\u1eb3ng.<\/li>\n<li>B\u1ea3o to\u00e0n t\u1ec9 s\u1ed1 \u0111\u1ed9 d\u00e0i c\u1ee7a hai \u0111o\u1ea1n th\u1eb3ng n\u1eb1m tr\u00ean hai \u0111\u01b0\u1eddng th\u1eb3ng song song ho\u1eb7c c\u00f9ng n\u1eb1m tr\u00ean m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng.<\/li>\n<\/ul>\n<\/li>\n<li><strong>H\u00ecnh bi\u1ec3u di\u1ec5n c\u1ee7a m\u1ed9t s\u1ed1 h\u00ecnh ph\u1eb3ng trong kh\u00f4ng gian:<\/strong>\n<ul>\n<li>H\u00ecnh bi\u1ec3u di\u1ec5n c\u1ee7a m\u1ed9t tam gi\u00e1c (\u0111\u1ec1u, vu\u00f4ng, c\u00e2n) b\u1ea5t k\u00ec l\u00e0 m\u1ed9t tam gi\u00e1c.<\/li>\n<li>H\u00ecnh bi\u1ec3u di\u1ec5n c\u1ee7a h\u00ecnh b\u00ecnh h\u00e0nh, h\u00ecnh ch\u1eef nh\u1eadt, h\u00ecnh vu\u00f4ng \u0111\u1ec1u l\u00e0 m\u1ed9t h\u00ecnh b\u00ecnh h\u00e0nh.<\/li>\n<li>H\u00ecnh bi\u1ec3u di\u1ec5n c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n l\u00e0 m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n ho\u1eb7c elip.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!-- H\u1ed8P L\u1ed6I SAI TH\u01af\u1edcNG G\u1eb6P --><\/p>\n<div style=\"background-color: #fff5f5; border-left: 4px solid #e53e3e; padding: 14px 18px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"margin: 0 0 6px 0; font-weight: bold; color: #c53030;\">\u26a0\ufe0f L\u1ed7i sai th\u01b0\u1eddng g\u1eb7p v\u1ec1 ph\u00e9p chi\u1ebfu song song:<\/p>\n<ul style=\"margin: 0; padding-left: 20px; font-size: 15px;\">\n<li>Nh\u1ea7m t\u01b0\u1edfng h\u00ecnh vu\u00f4ng trong kh\u00f4ng gian khi v\u1ebd h\u00ecnh bi\u1ec3u di\u1ec5n tr\u00ean m\u1eb7t ph\u1eb3ng v\u1eabn gi\u1eef nguy\u00ean g\u00f3c vu\u00f4ng v\u00e0 \u0111\u1ed9 d\u00e0i c\u00e1c c\u1ea1nh (trong khi h\u00ecnh bi\u1ec3u di\u1ec5n ch\u1ec9 l\u00e0 h\u00ecnh b\u00ecnh h\u00e0nh).<\/li>\n<\/ul>\n<\/div>\n<p><!-- V\u00cd D\u1ee4 M\u1eaaU --><\/p>\n<div style=\"background-color: #f7fafc; border-left: 4px solid #0056b3; padding: 16px 20px; border-radius: 4px; margin: 18px 0;\">\n<p style=\"font-weight: bold; color: #0056b3; margin: 0 0 8px 0;\">V\u00ed d\u1ee5 m\u1eabu gi\u1ea3i chi ti\u1ebft: \u1ee8ng d\u1ee5ng t\u00ednh ch\u1ea5t t\u1ec9 s\u1ed1 ph\u00e9p chi\u1ebfu<\/p>\n<p style=\"margin: 0 0 6px 0;\"><strong>\u0110\u1ec1 b\u00e0i:<\/strong> Cho \u0111o\u1ea1n th\u1eb3ng AB n\u1eb1m trong kh\u00f4ng gian. G\u1ecdi A&#8217; v\u00e0 B&#8217; l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu song song c\u1ee7a A v\u00e0 B tr\u00ean m\u1eb7t ph\u1eb3ng (P). Bi\u1ebft \u0111o\u1ea1n th\u1eb3ng AB c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng 10 cm v\u00e0 g\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. T\u00ednh \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng h\u00ecnh chi\u1ebfu M&#8217; c\u1ee7a M.<\/p>\n<p style=\"margin: 0;\"><strong>L\u1eddi gi\u1ea3i chi ti\u1ebft:<\/strong><br \/>\n\u2022 Theo t\u00ednh ch\u1ea5t b\u1ea3o to\u00e0n t\u1ec9 s\u1ed1 \u0111\u1ed9 d\u00e0i c\u1ee7a ph\u00e9p chi\u1ebfu song song:<br \/>\n&#8211; Trung \u0111i\u1ec3m M c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB s\u1ebd \u0111\u01b0\u1ee3c chi\u1ebfu th\u00e0nh trung \u0111i\u1ec3m M&#8217; c\u1ee7a \u0111o\u1ea1n th\u1eb3ng h\u00ecnh chi\u1ebfu A&#8217;B&#8217;.<br \/>\n\u2022 Do \u0111\u00f3 ta c\u00f3 h\u1ec7 th\u1ee9c t\u1ec9 l\u1ec7:<\/p>\n<p style=\"margin: 0;\"><code>A'M' \/ A'B' = AM \/ AB = 1\/2<\/code>.<br \/>\n\u2022 Suy ra \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng h\u00ecnh chi\u1ebfu M&#8217;B&#8217; (ho\u1eb7c A&#8217;M&#8217;) b\u1eb1ng m\u1ed9t n\u1eeda \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng h\u00ecnh chi\u1ebfu A&#8217;B&#8217;.<br \/>\n\u2022 <em>\u0110\u00e1p s\u1ed1:<\/em> \u0110o\u1ea1n th\u1eb3ng M&#8217; n\u1eb1m \u1edf v\u1ecb tr\u00ed <strong>chia \u0111\u00f4i \u0111o\u1ea1n h\u00ecnh chi\u1ebfu A&#8217;B&#8217; theo t\u1ec9 l\u1ec7 1 : 1<\/strong>.<\/p>\n<\/div>\n<p><!-- L\u01afU \u00dd \/ INTERNAL LINK BOX --><\/p>\n<div style=\"background-color: #edf2f7; border-left: 4px solid #0056b3; padding: 14px 18px; border-radius: 4px; margin-top: 25px;\">\n<p style=\"margin: 0; font-size: 15px; color: #2d3748;\"><strong>\ud83d\udca1 L\u1eddi khuy\u00ean t\u1eeb chuy\u00ean gia NaviEdu:<\/strong> H\u00ecnh h\u1ecdc kh\u00f4ng gian l\u1edbp 11 \u0111\u00f2i h\u1ecfi h\u1ecdc sinh ph\u1ea3i c\u00f3 t\u01b0 duy tr\u1ef1c quan v\u00e0 n\u1eafm ch\u1eafc c\u00e1c \u0111\u1ecbnh l\u00ed c\u01a1 b\u1ea3n. N\u1ebfu con \u0111ang c\u1ea3m th\u1ea5y b\u1ee1 ng\u1ee1 ho\u1eb7c m\u1ea5t g\u1ed1c h\u00ecnh h\u1ecdc kh\u00f4ng gian, ba m\u1eb9 h\u00e3y tham kh\u1ea3o ngay l\u1ed9 tr\u00ecnh <a style=\"color: #0056b3; font-weight: bold; text-decoration: underline;\" href=\"https:\/\/navi.edu.vn\/mat-goc-toan\/lop-11\">l\u1ea5y l\u1ea1i g\u1ed1c To\u00e1n 11<\/a> t\u1ea1i NaviEdu \u0111\u1ec3 h\u1ed7 tr\u1ee3 con t\u1ef1 tin chinh ph\u1ee5c \u0111i\u1ec3m s\u1ed1 cao!<\/p>\n<\/div>\n<\/div>\n<p><!-- K\u1ebeT TH\u00daC: PH\u1ea6N II - \u0110\u01af\u1edcNG TH\u1eb2NG V\u00c0 M\u1eb6T PH\u1eb2NG TRONG KH\u00d4NG GIAN TO\u00c1N 11 --><\/p>\n<p><!-- B\u1eaeT \u0110\u1ea6U PH\u1ea6N III: L\u1ecaCH \u00d4N T\u1eacP 7 NG\u00c0Y & \u0110\u1ec0 THI M\u1eaaU TO\u00c1N 11 --><\/p>\n<div class=\"naviedu-content-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<p><!-- BANNER PH\u1ea6N III --><\/p>\n<div style=\"background: linear-gradient(135deg, #0056b3, #0080ff); color: #ffffff; padding: 18px 24px; border-radius: 8px; margin-top: 35px; margin-bottom: 30px;\">\n<h2 id=\"muc-16-phan-iii-lich-on-tap-7-ngay-de-thi-mau\" style=\"color: #ffffff; margin: 0; font-size: 22px; text-transform: uppercase; letter-spacing: 0.5px;\">Ph\u1ea7n III: L\u1ecbch \u00d4n T\u1eadp 7 Ng\u00e0y &amp; \u0110\u1ec1 Thi M\u1eabu<\/h2>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 1: L\u1ecaCH \u00d4N T\u1eacP 7 NG\u00c0Y --><\/p>\n<p style=\"margin: 0 0 16px 0;\">Khi \u0111\u00e3 \u0111i h\u1ebft c\u00e1c ch\u1ee7 \u0111\u1ec1 trong \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11, con c\u00f3 th\u1ec3 d\u00f9ng l\u1ecbch 7 ng\u00e0y d\u01b0\u1edbi \u0111\u00e2y \u0111\u1ec3 h\u1ec7 th\u1ed1ng l\u1ea1i ki\u1ebfn th\u1ee9c v\u00e0 luy\u1ec7n \u0111\u1ec1 \u0111\u1ecbnh d\u1ea1ng m\u1edbi tr\u01b0\u1edbc ng\u00e0y thi.<\/p>\n<h2 id=\"muc-17-lich-on-tap-giua-ki-1-toan-11-trong-7-ngay-giup-but-pha-diem\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">L\u1ecbch \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 trong 7 ng\u00e0y gi\u00fap b\u1ee9t ph\u00e1 \u0111i\u1ec3m s\u1ed1<\/h2>\n<p>\u0110\u1ec3 gi\u00fap h\u1ecdc sinh THPT l\u00e0m ch\u1ee7 tr\u1ecdn v\u1eb9n kh\u1ed1i l\u01b0\u1ee3ng ki\u1ebfn th\u1ee9c tr\u1ecdng t\u00e2m m\u00f4n To\u00e1n l\u1edbp 11 v\u00e0 th\u00edch \u1ee9ng ho\u00e0n h\u1ea3o v\u1edbi \u0111\u1ecbnh d\u1ea1ng \u0111\u1ec1 thi m\u1edbi, \u0111\u1ed9i ng\u0169 chuy\u00ean m\u00f4n t\u1ea1i NaviEdu x\u00e2y d\u1ef1ng th\u1eddi gian bi\u1ec3u n\u01b0\u1edbc r\u00fat trong 7 ng\u00e0y (m\u1ed7i ng\u00e0y t\u1eeb 60 \u2013 90 ph\u00fat \u00f4n t\u1eadp chuy\u00ean s\u00e2u):<\/p>\n<div style=\"overflow-x: auto; margin: 20px 0;\">\n<table style=\"min-width: 640px; table-layout: fixed; font-size: 15px; width: 100%; border-collapse: collapse; border: 1px solid #cbd5e0; text-align: left;\">\n<colgroup>\n<col style=\"width: 16%;\" \/>\n<col style=\"width: 48%;\" \/>\n<col style=\"width: 36%;\" \/><\/colgroup>\n<thead style=\"background-color: #edf2f7;\">\n<tr>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; color: #0056b3; width: 18%;\">Th\u1eddi gian<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; color: #0056b3; width: 47%;\">N\u1ed9i dung \u00f4n<\/th>\n<th style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; color: #0056b3; width: 35%;\">M\u1ee5c ti\u00eau (60\u201390 ph\u00fat\/ng\u00e0y)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; font-weight: bold;\">Ng\u00e0y 1<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">G\u00f3c l\u01b0\u1ee3ng gi\u00e1c, c\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n, cung li\u00ean quan.<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">Th\u00e0nh th\u1ea1o \u0111\u1ed5i \u0111\u01a1n v\u1ecb radian, nh\u1edb chu\u1ea9n h\u1eb1ng \u0111\u1eb3ng th\u1ee9c v\u00e0 cung li\u00ean quan \u0111\u1eb7c bi\u1ec7t.<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; font-weight: bold;\">Ng\u00e0y 2<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">C\u00f4ng th\u1ee9c c\u1ed9ng, nh\u00e2n \u0111\u00f4i, bi\u1ebfn \u0111\u1ed5i t\u00edch th\u00e0nh t\u1ed5ng.<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">N\u1eafm ch\u1eafc k\u1ef9 n\u0103ng bi\u1ebfn \u0111\u1ed5i linh ho\u1ea1t c\u00e1c c\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c n\u00e2ng cao \u0111\u1ec3 r\u00fat g\u1ecdn bi\u1ec3u th\u1ee9c.<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; font-weight: bold;\">Ng\u00e0y 3<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">H\u00e0m s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c, t\u1eadp x\u00e1c \u0111\u1ecbnh, t\u00ednh tu\u1ea7n ho\u00e0n.<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">R\u00e8n luy\u1ec7n th\u00e0nh th\u1ea1o c\u00e1ch t\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh ch\u1ee9a ph\u00e2n th\u1ee9c\/c\u0103n v\u00e0 x\u00e1c \u0111\u1ecbnh chu k\u00ec h\u00e0m s\u1ed1 h\u1ee3p.<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; font-weight: bold;\">Ng\u00e0y 4<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">Ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n, GTLN \u2013 GTNN.<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">Gi\u1ea3i tr\u01a1n tru ph\u01b0\u01a1ng tr\u00ecnh c\u01a1 b\u1ea3n, \u0111\u1ebfm nghi\u1ec7m tr\u00ean kho\u1ea3ng v\u00e0 t\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t nh\u1ecf nh\u1ea5t.<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; font-weight: bold;\">Ng\u00e0y 5<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">H\u00ecnh h\u1ecdc kh\u00f4ng gian: x\u00e1c \u0111\u1ecbnh m\u1eb7t ph\u1eb3ng, giao tuy\u1ebfn, giao \u0111i\u1ec3m, ph\u00e9p chi\u1ebfu song song.<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">C\u1ee7ng c\u1ed1 t\u01b0 duy tr\u1ef1c quan kh\u00f4ng gian, th\u00e0nh th\u1ea1o t\u00ecm giao tuy\u1ebfn v\u00e0 giao \u0111i\u1ec3m tr\u00ean h\u00ecnh ch\u00f3p.<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; font-weight: bold;\">Ng\u00e0y 6<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">L\u00e0m 2 \u0111\u1ec1 m\u1eabu \u0111\u1ecbnh d\u1ea1ng m\u1edbi, b\u1ea5m gi\u1edd 90 ph\u00fat.<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">L\u00e0m quen \u00e1p l\u1ef1c ph\u00f2ng thi th\u1eadt v\u1edbi c\u1ea5u tr\u00fac 4 ph\u1ea7n, t\u1ed1i \u01b0u h\u00f3a th\u1eddi gian ph\u00e2n b\u1ed5 t\u1eebng ph\u1ea7n.<\/td>\n<\/tr>\n<tr>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0; font-weight: bold;\">Ng\u00e0y 7<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">Xem l\u1ea1i l\u1ed7i sai, \u0111\u1eb7c bi\u1ec7t c\u00e2u \u0111\u00fang\/sai v\u00e0 tr\u1ea3 l\u1eddi ng\u1eafn.<\/td>\n<td style=\"word-break: normal; overflow-wrap: normal; hyphens: none; padding: 12px 14px; border: 1px solid #cbd5e0;\">Kh\u1eafc ph\u1ee5c tri\u1ec7t \u0111\u1ec3 c\u00e1c b\u1eaby t\u01b0 duy \u1edf ph\u1ea7n \u0111\u00fang\/sai v\u00e0 r\u00e8n t\u00ednh c\u1ea9n th\u1eadn \u1edf ph\u1ea7n tr\u1ea3 l\u1eddi ng\u1eafn.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><!-- H\u1ed8P G\u1ee2I \u00dd \/ BLOG LINK --><\/p>\n<div style=\"background-color: #edf2f7; border-left: 4px solid #0056b3; padding: 14px 18px; border-radius: 4px; margin: 20px 0;\">\n<p style=\"margin: 0 0 10px 0; font-size: 15px; color: #2d3748;\"><strong>\ud83d\udca1 L\u1eddi khuy\u00ean cho h\u1ecdc sinh THPT:<\/strong> Giai \u0111o\u1ea1n n\u01b0\u1edbc r\u00fat l\u1edbp 11 \u0111\u00f2i h\u1ecfi ph\u01b0\u01a1ng ph\u00e1p t\u1ef1 h\u1ecdc khoa h\u1ecdc. Ba m\u1eb9 v\u00e0 h\u1ecdc sinh c\u00f3 th\u1ec3 tham kh\u1ea3o th\u00eam c\u00e1c ph\u01b0\u01a1ng ph\u00e1p <a style=\"color: #0056b3; font-weight: bold; text-decoration: underline;\" href=\"https:\/\/navi.edu.vn\/blog\/cach-hoc-toan-hieu-qua-thu-tu-hoc-chuan\/\">c\u00e1ch h\u1ecdc To\u00e1n hi\u1ec7u qu\u1ea3<\/a> \u0111\u1ec3 t\u1ed1i \u01b0u h\u00f3a n\u0103ng l\u1ef1c ti\u1ebfp thu m\u1ed7i ng\u00e0y.<\/p>\n<p style=\"margin: 0; font-size: 15px; color: #2d3748;\">B\u00ean c\u1ea1nh \u0111\u00f3, \u0111\u1ec3 c\u00f3 c\u00e1i nh\u00ecn t\u1ed5ng quan v\u1ec1 l\u1ed9 tr\u00ecnh \u00f4n thi THPT qu\u1ed1c gia s\u1eafp t\u1edbi, h\u1ecdc sinh c\u00f3 th\u1ec3 xem tr\u01b0\u1edbc <a style=\"color: #0056b3; font-weight: bold; text-decoration: underline;\" href=\"https:\/\/navi.edu.vn\/blog\/de-cuong-on-tap-giua-ki-1-toan-12-2026-2027\/\">\u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 12<\/a> nh\u1eb1m h\u00ecnh dung r\u00f5 h\u01a1n v\u1ec1 ch\u1eb7ng \u0111\u01b0\u1eddng chinh ph\u1ee5c \u0111\u1ec9nh cao to\u00e1n h\u1ecdc ti\u1ebfp theo!<\/p>\n<\/div>\n<p><!-- CH\u1ee6 \u0110\u1ec0 2: T\u1ea2I \u0110\u1ec0 C\u01af\u01a0NG V\u00c0 \u0110\u1ec0 THI M\u1eaaU --><\/p>\n<h2 id=\"muc-18-tai-de-cuong-on-tap-giua-ki-1-toan-11-va-de-thi-mau-pdf-co-d\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">T\u1ea3i \u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 v\u00e0 \u0111\u1ec1 thi m\u1eabu (PDF c\u00f3 \u0111\u00e1p \u00e1n)<\/h2>\n<p>Nh\u1eb1m gi\u00fap h\u1ecdc sinh l\u1edbp 11 c\u1ecd x\u00e1t v\u1edbi c\u1ea5u tr\u00fac \u0111\u1ec1 ki\u1ec3m tra \u0111\u1ecbnh k\u00ec theo \u0111\u1ecbnh d\u1ea1ng m\u1edbi nh\u1ea5t, \u0111\u1ed9i ng\u0169 gi\u00e1o vi\u00ean chuy\u00ean m\u00f4n t\u1ea1i NaviEdu \u0111\u00e3 bi\u00ean so\u1ea1n tr\u1ecdn b\u1ed9 t\u00e0i li\u1ec7u \u0111\u1ed9c quy\u1ec1n b\u00e1m s\u00e1t Th\u00f4ng t\u01b0 22:<\/p>\n<div style=\"background-color: #f7fafc; border: 1px solid #e2e8f0; border-radius: 8px; padding: 22px; margin: 20px 0;\">\n<h3 style=\"color: #2b6cb0; font-size: 17px; margin-top: 0; margin-bottom: 12px;\">T\u00f3m t\u1eaft c\u1ea5u tr\u00fac b\u1ed9 \u0111\u1ec1 m\u1eabu NaviEdu theo \u0111\u1ecbnh d\u1ea1ng m\u1edbi:<\/h3>\n<ul style=\"padding-left: 20px; margin-bottom: 15px;\">\n<li><strong>Ph\u1ea7n I \u2013 Tr\u1eafc nghi\u1ec7m nhi\u1ec1u l\u1ef1a ch\u1ecdn (3.0 \u0111i\u1ec3m):<\/strong> G\u1ed3m 12 c\u00e2u h\u1ecfi (m\u1ed7i c\u00e2u 0.25 \u0111i\u1ec3m), ki\u1ec3m tra nhanh nh\u1eadn bi\u1ebft v\u00e0 th\u00f4ng hi\u1ec3u to\u00e0n b\u1ed9 chuy\u00ean \u0111\u1ec1 l\u01b0\u1ee3ng gi\u00e1c v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian.<\/li>\n<li><strong>Ph\u1ea7n II \u2013 Tr\u1eafc nghi\u1ec7m \u0111\u00fang\/sai (2.0 \u0111i\u1ec3m):<\/strong> G\u1ed3m 2 c\u00e2u h\u1ecfi l\u1edbn d\u1ea1ng chu\u1ed7i \u00fd a, b, c, d (ch\u1ea5m \u0111i\u1ec3m l\u0169y ti\u1ebfn t\u1ed1i \u0111a 1 \u0111i\u1ec3m\/c\u00e2u), \u0111\u00e1nh gi\u00e1 s\u00e2u t\u01b0 duy logic \u0111a chi\u1ec1u c\u1ee7a h\u1ecdc sinh.<\/li>\n<li><strong>Ph\u1ea7n III \u2013 Tr\u1eafc nghi\u1ec7m tr\u1ea3 l\u1eddi ng\u1eafn (2.0 \u0111i\u1ec3m):<\/strong> G\u1ed3m 4 c\u00e2u h\u1ecfi y\u00eau c\u1ea7u h\u1ecdc sinh t\u1ef1 t\u00ednh to\u00e1n v\u00e0 \u0111i\u1ec1n \u0111\u00e1p s\u1ed1 ch\u00ednh x\u00e1c, th\u1eed th\u00e1ch n\u0103ng l\u1ef1c bi\u1ebfn \u0111\u1ed5i bi\u1ec3u th\u1ee9c v\u00e0 x\u00e1c \u0111\u1ecbnh giao tuy\u1ebfn.<\/li>\n<li><strong>Ph\u1ea7n IV \u2013 T\u1ef1 lu\u1eadn (3.0 \u0111i\u1ec3m):<\/strong> G\u1ed3m c\u00e1c b\u00e0i to\u00e1n t\u1ef1 lu\u1eadn chuy\u00ean s\u00e2u v\u1ec1 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh l\u01b0\u1ee3ng gi\u00e1c, t\u00ecm GTLN-GTNN v\u00e0 b\u00e0i to\u00e1n ch\u1ee9ng minh h\u00ecnh h\u1ecdc kh\u00f4ng gian.<\/li>\n<\/ul>\n<p style=\"margin-top: 15px; margin-bottom: 20px; color: #4a5568;\">To\u00e0n b\u1ed9 t\u00e0i li\u1ec7u bao g\u1ed3m t\u00f3m t\u1eaft l\u00fd thuy\u1ebft tr\u1ecdng t\u00e2m, h\u1ec7 th\u1ed1ng b\u00e0i t\u1eadp t\u1ef1 luy\u1ec7n theo t\u1eebng chuy\u00ean \u0111\u1ec1 v\u00e0 03 \u0111\u1ec1 thi th\u1eed chu\u1ea9n format 90 ph\u00fat c\u00f3 k\u00e8m b\u1ea3ng \u0111\u00e1p \u00e1n, h\u01b0\u1edbng d\u1eabn gi\u1ea3i chi ti\u1ebft t\u1eebng b\u01b0\u1edbc gi\u00fap con t\u1ef1 h\u1ecdc hi\u1ec7u qu\u1ea3 t\u1ea1i nh\u00e0:<\/p>\n<div style=\"text-align: center; margin-top: 25px; padding-top: 18px; border-top: 1px dashed #cbd5e0;\"><a style=\"background-color: #d9534f; color: #ffffff; padding: 13px 30px; text-decoration: none; border-radius: 6px; font-weight: bold; font-size: 16px; display: inline-block; box-shadow: 0 2px 5px rgba(217,83,79,0.3);\" href=\"https:\/\/navi.edu.vn\/thu-vien\/danh-muc-tai-lieu\/lop-11\/\">\ud83d\udce5 T\u1ea2I TR\u1eccN B\u1ed8 \u0110\u1ec0 C\u01af\u01a0NG &amp; \u0110\u1ec0 M\u1eaaU TO\u00c1N 11 (PDF)<br \/>\n<\/a><\/p>\n<p style=\"margin-top: 12px; font-size: 14px; color: #4a5568;\">(T\u00e0i li\u1ec7u \u0111\u01b0\u1ee3c c\u1eadp nh\u1eadt mi\u1ec5n ph\u00ed t\u1ea1i kho <a style=\"color: #0056b3; font-weight: bold; text-decoration: underline;\" href=\"https:\/\/navi.edu.vn\/thu-vien\/danh-muc-tai-lieu\/lop-11\/\">th\u01b0 vi\u1ec7n h\u1ecdc li\u1ec7u NaviEdu<\/a>)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><!-- K\u1ebeT TH\u00daC PH\u1ea6N III: L\u1ecaCH \u00d4N T\u1eacP 7 NG\u00c0Y & \u0110\u1ec0 THI M\u1eaaU TO\u00c1N 11 --><\/p>\n<p><!-- B\u1eaeT \u0110\u1ea6U K\u1ebeT B\u00c0I V\u00c0 KH\u1ed0I CTA WORDPRESS TO\u00c1N 11 --><\/p>\n<div class=\"naviedu-content-section\" style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif; line-height: 1.8; color: #2d3748; font-size: 16px;\">\n<h2 id=\"muc-19-loi-ket\" style=\"color: #0056b3; border-bottom: 2px solid #e2e8f0; padding-bottom: 8px; margin-top: 35px; font-size: 20px;\">L\u1eddi k\u1ebft<\/h2>\n<p>Chinh ph\u1ee5c ch\u01b0\u01a1ng tr\u00ecnh To\u00e1n 11 v\u1edbi l\u01b0\u1ee3ng c\u00f4ng th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c \u0111\u1ed3 s\u1ed9 v\u00e0 t\u01b0 duy h\u00ecnh h\u1ecdc kh\u00f4ng gian tr\u1eebu t\u01b0\u1ee3ng \u0111\u00f2i h\u1ecfi c\u00e1c em h\u1ecdc sinh s\u1ef1 ki\u00ean tr\u00ec c\u00f9ng ph\u01b0\u01a1ng ph\u00e1p \u00f4n luy\u1ec7n b\u00e0i b\u1ea3n. Vi\u1ec7c n\u1eafm v\u1eefng c\u1ea5u tr\u00fac \u0111\u1ec1 ki\u1ec3m tra v\u00e0 ch\u1ee7 \u0111\u1ed9ng luy\u1ec7n gi\u1ea3i c\u00e1c d\u1ea1ng b\u00e0i tr\u1ecdng t\u00e2m ch\u00ednh l\u00e0 ch\u00eca kh\u00f3a v\u00e0ng gi\u00fap c\u00e1c em t\u1ef1 tin b\u01b0\u1edbc v\u00e0o ph\u00f2ng thi.<\/p>\n<p>Hy v\u1ecdng r\u1eb1ng b\u1ed9 <strong>\u0111\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11<\/strong> do \u0111\u1ed9i ng\u0169 chuy\u00ean m\u00f4n NaviEdu bi\u00ean so\u1ea1n s\u1ebd l\u00e0 ng\u01b0\u1eddi b\u1ea1n \u0111\u1ed3ng h\u00e0nh tin c\u1eady, gi\u00fap ba m\u1eb9 v\u00e0 c\u00e1c con g\u1ee1 r\u1ed1i m\u1ecdi kh\u00f3 kh\u0103n. Ch\u00fac c\u00e1c em gi\u1eef v\u1eefng tinh th\u1ea7n t\u1eadp trung cao \u0111\u1ed9, \u00f4n t\u1eadp th\u00f4ng minh v\u00e0 gi\u00e0nh tr\u1ecdn \u0111i\u1ec3m s\u1ed1 th\u1eadt xu\u1ea5t s\u1eafc trong k\u00ec thi s\u1eafp t\u1edbi!<\/p>\n<p><!-- KH\u1ed0I CTA WORDPRESS CHU\u1ea8N CHUY\u1ec2N \u0110\u1ed4I --><\/p>\n<div style=\"border: 2px dashed #0056b3; background-color: #f4f8ff; border-radius: 10px; padding: 25px; text-align: center; margin: 35px 0;\">\n<h3 style=\"color: #0056b3; margin-top: 0; font-size: 20px; font-weight: bold;\">B\u1ee9t Ph\u00e1 \u0110i\u1ec3m S\u1ed1 Gi\u1eefa K\u00ec 1 To\u00e1n 11 C\u00f9ng L\u1ed9 Tr\u00ecnh C\u00e1 Nh\u00e2n H\u00f3a T\u1ea1i NaviEdu!<\/h3>\n<p style=\"font-size: 16px; line-height: 1.6; color: #333333; margin-bottom: 20px; max-width: 750px; margin-left: auto; margin-right: auto;\">H\u1ecdc sinh v\u00e0 ba m\u1eb9 h\u00e3y \u0111\u01b0a con \u0111\u1ebfn c\u01a1 s\u1edf <strong>NaviEdu (33 LK6B M\u1ed7 Lao, L\u00e0ng Vi\u1ec7t Ki\u1ec1u Ch\u00e2u \u00c2u, H\u00e0 \u0110\u00f4ng)<\/strong> \u0111\u1ec3 tham gia l\u00e0m b\u00e0i \u0111\u00e1nh gi\u00e1 n\u0103ng l\u1ef1c mi\u1ec5n ph\u00ed b\u1eb1ng Math AI v\u00e0 nh\u1eadn l\u1ed9 tr\u00ecnh h\u1ecdc &#8220;may \u0111o&#8221; ri\u00eang bi\u1ec7t cho ch\u00ednh m\u00ecnh ngay h\u00f4m nay!<\/p>\n<div><a style=\"background-color: #0056b3; color: #ffffff; padding: 13px 30px; text-decoration: none; border-radius: 6px; font-weight: bold; font-size: 16px; display: inline-block; box-shadow: 0 2px 4px rgba(0,86,179,0.2);\" href=\"https:\/\/navi.edu.vn\/giai-phap\/dang-ky-tu-van?utm_campaign=All.Blog._Organic.Dieuttm_Testtoan11.cta.2610_._.&amp;utm_medium=Dieuttm\" target=\"_blank\" rel=\"noopener\">L\u00c0M B\u00c0I \u0110\u00c1NH GI\u00c1 N\u0102NG L\u1ef0C MI\u1ec4N PH\u00cd<\/a><\/div>\n<\/div>\n<\/div>\n<p><!-- K\u1ebeT TH\u00daC K\u1ebeT B\u00c0I V\u00c0 KH\u1ed0I CTA WORDPRESS TO\u00c1N 11 --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0110\u1ec1 c\u01b0\u01a1ng \u00f4n t\u1eadp gi\u1eefa k\u00ec 1 To\u00e1n 11 (2026\u20132027): ki\u1ebfn th\u1ee9c tr\u1ecdng t\u00e2m, d\u1ea1ng b\u00e0i hay g\u1eb7p, \u0111\u1ec1 m\u1eabu c\u00f3 \u0111\u00e1p \u00e1n, l\u1ecbch \u00f4n 7 ng\u00e0y v\u00e0 file PDF t\u1ea3i mi\u1ec5n ph\u00ed.<\/p>\n","protected":false},"author":2,"featured_media":2912,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[6],"tags":[],"_links":{"self":[{"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/posts\/2869"}],"collection":[{"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/comments?post=2869"}],"version-history":[{"count":6,"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/posts\/2869\/revisions"}],"predecessor-version":[{"id":2915,"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/posts\/2869\/revisions\/2915"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/media\/2912"}],"wp:attachment":[{"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/media?parent=2869"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/categories?post=2869"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/navi.edu.vn\/blog\/wp-json\/wp\/v2\/tags?post=2869"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}