{"id":3178,"date":"2024-10-25T17:02:53","date_gmt":"2024-10-25T09:02:53","guid":{"rendered":"http:\/\/jhwk.online\/?p=3178"},"modified":"2024-10-25T17:02:53","modified_gmt":"2024-10-25T09:02:53","slug":"hsyldwt-5-3-4-5-2-2","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=3178","title":{"rendered":"\u6b63\u5f26\u5b9a\u7406\u548c\u4f59\u5f26\u5b9a\u7406"},"content":{"rendered":"\n<div class=\"wp-block-ht-block-toc  is-style-outline htoc htoc--position-wide toc-list-style-plain\" data-htoc-state=\"expanded\"><span class=\"htoc__title\"><span class=\"ht_toc_title\">\u76ee\u5f55<\/span><span class=\"htoc__toggle\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\"><g fill=\"#444\"><path d=\"M15 7H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><path d=\"M15 1H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1zM15 13H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><\/g><\/svg><\/span><\/span><div class=\"htoc__itemswrap\"><ul class=\"ht_toc_list\"><li class=\"\"><a href=\"\/#htoc-\">\u6b63\u5f26\u5b9a\u7406\u3001\u4f59\u5f26\u5b9a\u7406\u6a21\u578b<\/a><\/li><li class=\"\"><a href=\"\/#htoc-1111111111\">\u6c42\u6700\u503c\u6216\u8303\u56f4<\/a><\/li><li class=\"\"><a href=\"\/#htoc-111111111111111\">\u4e09\u89d2\u5f62\u7684\u9ad8\u7ebf\u3001\u4e2d\u7ebf\u3001\u89d2\u5e73\u5206\u7ebf<\/a><\/li><\/ul><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-\">\u6b63\u5f26\u5b9a\u7406\u3001\u4f59\u5f26\u5b9a\u7406\u6a21\u578b<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.(\u591a\u9009)\u4e0b\u5217\u6b63\u786e\u7684\u9009\u9879\u662f(\\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>A.\u5728  \\triangle A B C  \u4e2d, \u82e5  \\sin A&gt;\\sin B , \u5219  A&gt;B ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB.\u5728  \\triangle A B C  \u4e2d, \u82e5  \\sin 2 A=\\sin 2 B , \u5219  A=B ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nC.\u5f53  b^{2}+c^{2}-a^{2}&gt;0  \u65f6,  \\triangle A B C  \u4e3a\u9510\u89d2\u4e09\u89d2\u5f62; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\D.\u5f53  b^{2}+c^{2}-a^{2}&lt;0  \u65f6,  \\triangle A B C  \u4e3a\u949d\u89d2\u4e09\n\u89d2\u5f62;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nE.\u5728\u9510\u89d2\\triangle A B C\u4e2d\uff0c\\sin A&gt;\\cos B.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-ade-a-a-gt-b-leftrightarrow-a-gt-b-leftrightarrow-sin-a-gt-sin-b-leftrightarrow-cos-a-lt-cos-b-b-sin-2-a-sin-2-b-a-b-2a-2b-pi-e-a-b-gt-frac-pi-2-a-gt-frac-pi-2-b-sin-a-gt-sin-frac-pi-2-b\"><pre>\u7b54\u6848:ADE;\u63d0\u793a:A\u9009\u9879\u7684\u7b49\u4ef7\u5173\u7cfb\u6709:A&gt;B\\Leftrightarrow a&gt;b\\Leftrightarrow \\sin A &gt;\\sin B~~~~~~~~~~~~~~~~~~~~\\\\\\Leftrightarrow \\cos A&lt; \\cos B;\u9009\u9879B\u4e2d  \\sin 2 A=\\sin 2 B , \u5219  A=B \u62162A+2B=\\pi;\u9009\u9879E\u4e2d\uff0c~~~~~~~~~\\\\A+B&gt;\\frac{\\pi}{2},\u5219A&gt;\\frac{\\pi}{2}-B,\\sin A&gt;\\sin (\\frac{\\pi}{2}-B).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.(1)\u5728  \\triangle A B C  \u4e2d, \u5df2\u77e5  A C=1, B C=\\sqrt{3}, B=30^{\\circ} , \u5219  A=(\\quad) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  60^{\\circ} ~~~~\nB.  120^{\\circ} ~~~~\nC.  60^{\\circ}  \u6216  120^{\\circ} ~~~~\nD.  30^{\\circ}  \u6216  90^{\\circ} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5728  \\triangle A B C  \u4e2d, \u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9\u5206\u522b\u4e3a  a, b, c . \u82e5  a=2   ,b=\\sqrt{6}, B=\\frac{\\pi}{3} , \u5219A\u4e3a(\\quad)~~~~\\\\\nA.  \\frac{\\pi}{4} \\quad\nB.  \\frac{\\pi}{3} \\quad\nC.  \\frac{\\pi}{4}  \u6216  \\frac{3 \\pi}{4} \\quad\nD.  \\frac{\\pi}{3}  \u6216  \\frac{2 \\pi}{3} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-c-2-a-1-2-pi-3\"><pre>\u7b54\u6848(1)C,(2)A;\u63d0\u793a\uff1a\u2605\u5224\u65ad\u89e3\u7684\u4e2a\u6570\u7684\u65b9\u6cd5:1.\u5927\u8fb9\u5bf9\u5927\u89d2.2.\u4e09\u89d2\u5f62\u4e24\u89d2\u4e4b\u548c~~~~~~~~~\\\\\u5c0f\u4e8e\\pi.3.\u5df2\u77e5\u4e24\u8fb9\u53ca\u4e00\u8fb9\u7684\u5bf9\u89d2\u65f6\uff0c\u753b\u56fe\u4e0e\u9ad8\u6bd4\u8f83\u5927\u5c0f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\n(\u591a\u9009)\u5728  \\triangle A B C  \u4e2d, \u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9\u5206\u522b\u4e3a  a, b, c , \u6839\u636e\u4e0b\u5217\u6761\u4ef6\u89e3\u4e09\u89d2\u5f62, \u5176\u4e2d~~~~~~\\\\\u6709\u4e24\u89e3\u7684\u662f (\\quad )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  b=10, A=45^{\\circ}, C=60^{\\circ} \\quad\nB.  b=\\sqrt{15}, c=4, B=60^{\\circ} \\quad~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nC.  a=\\sqrt{3}, b=2, A=45^{\\circ} \\quad~~~\nD.  a=8, b=4, A=80^{\\circ} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-334d50e\" data-block-id=\"334d50e\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-334d50e-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-cc9fcc6\" data-v=\"4\" data-block-id=\"cc9fcc6\"><style>@media screen and (min-width:690px){.stk-cc9fcc6 {flex:var(--stk-flex-grow, 1) 1 66.7\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-bc\"><pre>\u7b54\u6848:BC;\u63d0\u793a\uff1a\u5df2\u77e5\u4e24\u8fb9\u53ca\u4e00\u8fb9\u7684\u5bf9\u89d2\uff0c\u53ef\u80fd\u4f1a\u6709\\\\\u4e24\u89e3\uff0c\u6839\u636e\u5df2\u77e5\u89d2\u505a\u9ad8\u7ebf.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-8cf56e9\" data-v=\"4\" data-block-id=\"8cf56e9\"><style>@media screen and (min-width:690px){.stk-8cf56e9 {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/10\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-10-25-200347.png\" alt=\"\" class=\"wp-image-3198\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.(1)\u5728  \\triangle A B C  \u4e2d, \u5185\u89d2  A, B, C  \u6240\u5bf9\u5e94\u7684\u8fb9\u5206\u522b\u662f  a, b, c ,\u82e5  a=3, b=\\sqrt{13}, B=60^{\\circ} , \u6c42 c.<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>(2)\u5728  \\triangle A B C  \u4e2d\uff0c  a=3, b=2 \\sqrt{6}, \\angle B=2 \\angle A .\n(1) \u6c42  \\cos A  \u7684\u503c;\n(2) \u6c42  c  \u7684\u503c.~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-4-b-2-a-2-c-2-2-a-c-cos-b-c-1-c-4\"><pre>\u7b54\u6848:(1)4;\u63d0\u793a: \u7531\u4f59\u5f26\u5b9a\u7406\u5f97  b^{2}=a^{2}+c^{2}-2 a c \\cos B , \u89e3\u5f97  c=-1  (\u820d\u53bb)\u6216  c=4 .~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1111111111111111111111\"><pre>(2)\\frac{\\sqrt{6}}{3},5;\u63d0\u793a: \\frac{3}{\\sin A}=\\frac{2 \\sqrt{6}}{\\sin 2 A} \\Rightarrow   \n \\cos A=\\frac{\\sqrt{6}}{3} ,\\Rightarrow   \\sin A=\\frac{\\sqrt{3}}{3} , \\because   \\angle B=2 \\angle A ,\n~~~~~~~~~~~~~\\\\\n\\Rightarrow   \\cos B=\\frac{1}{3} , \\sin B=\\frac{2 \\sqrt{2}}{3} .\n\u5728  \\triangle A B C  \u4e2d,  \\sin C=\\sin (A+B)=\\frac{5 \\sqrt{3}}{9} . \u7531\\frac{c}{\\sin C}=\\frac{a}{\\sin A}  \\\\\\Rightarrow   c=\\frac{a \\sin C}{\\sin A}=5 ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-zhu\"><pre>\u2605\u2605\u6ce8:\u4f59\u5f26\u5b9a\u7406\u7684\u89e3\u4e0d\u6613\u53d6\u820d\uff01\u7531a^2=b^2+c^2-2bc\\cos A \\Rightarrow c=3\u62165\uff0c\u54ea\u4e2a\u89e3\u4e0d~~~~\\\\\u6210\u7acb\u5462\uff1f\u8ba1\u7b97\\sin A=\\frac{3\\sqrt{3}}{9},\\sin B=\\frac{6\\sqrt{2}}{9},\\sin C=\\frac{5\\sqrt{3}}{9},\\therefore \\sin C&gt; \\sin B&gt; \\sin A, \\Rightarrow~~~~\\\\c&gt;b&gt;a, \\Rightarrow a=5.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5. \u5df2\u77e5  \\triangle A B C  \u7684\u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9\u5206\u522b\u662f  a, b, c , \u82e5  a=4, b=6, B=2 A , \u5219  c=(\\quad) \n~~~~~\\\\\nA. 5\\quad\nB. 4 \u6216 5\\quad\nC. 6\\quad\nD. 4 \u6216 6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-frac-a-sin-a-frac-b-sin-b-cos-a-frac-3-4-cos-b-cos-2-a-frac-1-8-sin-b-frac-3-sqrt-7-8-cos-c-cos-a-b-frac-9-16-c-2-a-2-b-2-2-a-b-cdot-cos-c-25-c-5-a\"><pre>\u7b54\u6848:A;\u63d0\u793a:\\frac{a}{\\sin A}=\\frac{b}{\\sin 2A} ,\u89e3\u5f97  \\cos A=\\frac{3}{4} ,  \\Rightarrow  \\cos B=\\cos 2 A=\\frac{1}{8} ,\n\\sin B=\\frac{3 \\sqrt{7}}{8} ,\\\\\n \\cos C=-\\cos (A+B)=\\frac{9}{16} ,\n c^{2}=a^{2}+b^{2}-2 a b \\cdot \\cos C=25 , \u89e3\u5f97  c=5 .\n\u6545\u9009:  A .~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u5728  \\triangle A B C  \u4e2d, \u82e5  2 \\cos ^{2} A-\\cos A=2 \\cos ^{2} B+2 \\cos ^{2} C-2+\\cos (B   -C ) \uff0c\u5219  A=(~~~) \\\\\nA.  \\frac{\\pi}{6} \\quad\nB.  \\frac{\\pi}{3} \\quad\nC.  \\frac{2 \\pi}{3} \\quad\nD.  \\frac{5 \\pi}{6} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-b-2-left-1-sin-2-a-right-cos-pi-b-c-2-left-1-sin-2-b-right-2-left-1-sin-2-c-right-2-cos-b-c-sin-2-b-sin-2-c-sin-2-a-sin-b-sin-c-rightarrow-b-2-c-2-a-2-b-c-cos-a-frac-1-2-a-in-0-pi-a-frac-pi-3\"><pre>\u7b54\u6848:B;\u63d0\u793a:\u7531\u5df2\u77e5\u5f97 2\\left(1-\\sin ^{2} A\\right)-\\cos [\\pi-(B+C)]=2\\left(1-\\sin ^{2} B\\right)+~~~~~~~~~~~~~~\\\\2\\left(1-\\sin ^{2} C\\right)-2+\\cos (B-C) ,\n\u6574\u7406\u5f97  \\sin ^{2} B+\\sin ^{2} C-\\sin ^{2} A=\\sin B \\sin C ,~~~~~~~~~~~~~\\\\\n\\Rightarrow  b^{2}+c^{2}-a^{2}=b c ,\n\u518d\u7531\u4f59\u5f26\u5b9a\u7406\u5f97  \\cos A=\\frac{1}{2} .\n\u56e0\u4e3a  A \\in(0, \\pi) , \u6545  A=\\frac{\\pi}{3} .~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7.\u8bb0  \\triangle A B C  \u7684\u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9\u5206\u522b\u4e3a  a, b, c , \u5df2\u77e5  \\frac{\\cos A}{1+\\sin A}=\\frac{\\sin 2 B}{1+\\cos 2 B},\n\u82e5  C=\\frac{2 \\pi}{3} ,\\\\ \u6c42  B .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-pi-6-rightarrow-frac-cos-a-1-sin-a-frac-2-sin-b-cos-b-2-cos-2-b-rightarrow-cos-a-b-sin-b-gt-0-therefore-a-b-therefore-cos-a-b-sin-left-frac-pi-2-a-b-right-sin-b-therefore-frac-pi-2-a-b-b-a-frac-pi-2-2-b-rightarrow-b-frac-pi-6\"><pre>\u7b54\u6848:\\frac{\\pi}{6};\u63d0\u793a:   \\Rightarrow \\frac{\\cos A}{1+\\sin A}=\\frac{2\\sin B\\cos B}{2\\cos ^2 B}\\Rightarrow\\cos (A+B)=\\sin B&gt;0 ,\\therefore A+B\u4e3a~~~~\\\\\u9510\u89d2, \\therefore \\cos(A+B)=  \\sin \\left[\\frac{\\pi}{2}-(A+B)\\right]=\\sin B ,  \\therefore  \\frac{\\pi}{2}-(A+B)=B , \u5373 A=\\frac{\\pi}{2}-2 B ,\\\\\\Rightarrow B=\\frac{\\pi}{6}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>8. \\triangle A B C  \u4e2d\u82e5\u6709  \\sin C=\\frac{\\sin A+\\sin B}{\\cos A+\\cos B} ,   \\triangle A B C  \u7684\u5f62\u72b6\u4e00\u5b9a\u662f ( \\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\A.\u9510\u89d2\u4e09\u89d2\u5f62\\quad B.\u949d\u89d2\u4e09\u89d2\u5f62\\quad C.\u76f4\u89d2\u4e09\u89d2\u5f62\\quad D.\u7b49\u8170\u76f4\u89d2\u4e09\u89d2\u5f62\\quad~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-b-sin-c-cos-a-cos-b-sin-a-sin-b-sin-b-c-sin-a-c-rightarrow-sin-b-cos-c-sin-a-cos-c-0-rightarrow-sin-b-sin-a-cos-c-0-rightarrow-c-frac-pi-2-rightarrow-triangle-a-b-c\"><pre>\u7b54\u6848:C;\u63d0\u793a:\\sin C(\\cos A+\\cos B)=\\sin A+\\sin B=\\sin (B+C)+\\sin (A+C),~~~~~~~~\\\\\\Rightarrow \\sin B \\cos C+\\sin A \\cos C=0, \\Rightarrow(\\sin B+\\sin A) \\cos C=0, \\Rightarrow C=\\frac{\\pi}{2}, \\Rightarrow \\triangle A B C~~~\\\\\u4e3a\u76f4\u89d2\u4e09\u89d2\u5f62.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>9.\u5728\u659c  \\triangle A B C  \u4e2d, \u82e5  \\sin A=\\cos B , \u5219  3 \\tan B+\\tan C  \u7684\u6700\u5c0f\u503c\u4e3a ( \\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  \\sqrt{2} \\quad\nB.  \\sqrt{5} \\quad\nC.  \\sqrt{6} \\quad\nD.  4 \\sqrt{3} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-b-sin-a-cos-b-gt-0-b-a-a-a-b-frac-pi-2-triangle-a-a-frac-pi-2-b-c-frac-pi-2-2-b-3-tan-b-tan-c-3-tan-b-tan-left-frac-pi-2-2-b-right-3-tan-b-frac-1-tan-2-b-3-tan-b-frac-1-tan-2-b-2-tan-b-frac-5-2-tan-b-frac-1-2-tan-b-geq-sqrt-5\"><pre>\u7b54\u6848:B;\u63d0\u793a:\n\u56e0\u4e3a \\sin A=\\cos B&gt;0 , \u6240\u4ee5  B  \u4e3a\u9510\u89d2,  A  \u53ef\u80fd\u4e3a\u9510\u89d2\u6216\u949d\u89d2.~~~~~~~~~~~~~~~~~~\\\\\n\u82e5  A  \u4e3a\u9510\u89d2  A+B=\\frac{\\pi}{2} , \u4e0e\u659c  \\triangle  \u77db\u76fe;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u82e5  A  \u4e3a\u949d\u89d2,  A=\\frac{\\pi}{2}+B, C=\\frac{\\pi}{2}-2 B ,3 \\tan B+\\tan C=3 \\tan B+\\tan \\left(\\frac{\\pi}{2}-2 B\\right)~~~~~~~~\\\\=3 \\tan B+\\frac{1}{\\tan 2 B} \n=3 \\tan B+\\frac{1-\\tan ^{2} B}{2 \\tan B}=\\frac{5}{2} \\tan B+\\frac{1}{2 \\tan B} \\geq \\sqrt{5}~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-1111111111\">\u6c42\u6700\u503c\u6216\u8303\u56f4<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\u5728  \\triangle A B C  \u4e2d, \u5185\u89d2  A, B, C  \u6240\u5bf9\u7684\u8fb9\u5206\u522b\u4e3a  a, b, c , \u4e14  a \\sin A+b \\sin B-a \\sin B=c \\sin C .\\\\\n(1) \u6c42\u89d2  C  \u7684\u503c\uff1b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) \u82e5  c=2 , \u6c42  \\triangle A B C  \u5468\u957f\u7684\u6700\u5927\u503c;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(3)\u82e5  c=2 , \u6c42  \\triangle A B C  \u9762\u79ef\u7684\u6700\u5927\u503c;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(4)\u82e5  c=\\sqrt{3}, \u6c42  \\triangle A B C  \u5468\u957f\u7684\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-frac-pi-3-2-6-3-sqrt-3-4-2-sqrt-3-3-sqrt-3-1-rightarrow-c-2-a-2-b-2-a-b-cos-c-frac-1-2-c-in-0-pi-c-frac-pi-3-2-a-2-b-2-a-b-c-2-4-4-a-b-2-3-a-b-ab-le-frac-a-b-2-2-a-b-leq-4-a-b-2-triangle-a-b-c-6-3-ab-le-4-s-frac-1-2-ab-sin-c-sqrt-3-4-frac-a-sin-a-frac-b-sin-b-frac-c-sin-c-2-rightarrow-b-2-sin-a-sin-b-2-sin-a-sin-frac-2-pi-3-a-2-sqrt-3-sin-a-frac-pi-6-a-in-0-frac-2-pi-3\"><pre>\u7b54\u6848:(1)\\frac{\\pi}{3},(2)6,(3)\\sqrt{3};(4)(2\\sqrt{3},3\\sqrt{3}]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u63d0\u793a\uff1a(1)\u539f\u5f0f\\Rightarrow   c^{2}=a^{2}+b^{2}-a b ,\n\u6545  \\cos C=\\frac{1}{2} ,   C \\in(0, \\pi) ,\n\u6545  C=\\frac{\\pi}{3} ;~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) \u7531  a^{2}+b^{2}-a b=c^{2}=4 ,\n\n\u5373  4=(a+b)^{2}-3 a b ;\u7531\u57fa\u672c\u4e0d\u7b49\u5f0fab \\le (\\frac{a+b}{2})^2~~~~~~~~~~~~~~~~\\\\\n\u5f97  a+b \\leq 4 ,\n\u5f53\u4e14\u4ec5\u5f53  a=b=2  \u65f6\u53d6\u5f97\u7b49\u53f7,\n\u6545  \\triangle A B C  \u5468\u957f\u7684\u6700\u5927\u503c\u4e3a 6 ;~~~~~~~~~~~~~~~~~~~~~~~~\\\\(3)\u540c\u7406ab\\le 4,\u9762\u79efS=\\frac{1}{2}ab\\sin C\u6700\u5927\u503c\u4e3a\\sqrt{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(4)\u8f6c\u5316\u4e3a\u51fd\u6570\u6c42\u8303\u56f4.\u7531\\frac{a}{\\sin A} =\\frac{b}{\\sin B}= \\frac{c}{\\sin C} =2,\\Rightarrow b=2(\\sin A+\\sin B)=~~~~~~~~~~~~~\\\\2(\\sin A+\\sin (\\frac{2\\pi}{3}-A)=2\\sqrt{3}\\sin (A+\\frac{\\pi }{6} )  ,\u5176\u4e2dA\\in (0,\\frac{2\\pi }{3}).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u5728\u9510\u89d2\u4e09\u89d2\u5f62  A B C  \u4e2d, \u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9\u5206\u522b\u4e3a  a, b, c , \u4e14  \\frac{\\cos A}{a}+\\frac{\\cos B}{b}=   \\frac{2 \\sqrt{3} \\sin C}{3 a} .\\\\\n(1) \u6c42\u89d2  B  \u7684\u5927\u5c0f;\n(2) \u82e5  b=2 \\sqrt{3} , \u6c42  a+c  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-frac-pi-3-2-6-4-sqrt-3-1-b-cos-a-a-cos-b-frac-2-sqrt-3-3-b-sin-c-sin-b-frac-sqrt-3-2-b-b-frac-pi-3-2-frac-a-sin-a-frac-c-sin-c-frac-b-sin-b-4-a-4-sin-a-c-4-sin-c-a-c-4-sin-a-4-sin-c-4-sin-a-4-sin-left-frac-2-pi-3-a-right-4-sqrt-3-sin-left-a-frac-pi-6-right-a-b-c-left-begin-matrix-0-a-displaystyle-frac-pi-2-0-displaystyle-frac-2-pi-3-a-displaystyle-frac-pi-2-end-matrix-right-frac-pi-6-lt-a-lt-frac-pi-2-therefore-frac-pi-3-lt-a-frac-pi-6-lt-frac-2-pi-3-therefore-frac-sqrt-3-2-lt-sin-left-a-frac-pi-6-right-leqslant-1-therefore-6-lt-a-c-leqslant-4-sqrt-3-therefore-a-c-6-4-sqrt-3\"><pre>\u7b54\u6848:(1)\\frac{\\pi}{3},(2)(6,4 \\sqrt{3}] ;\n\n\u63d0\u793a: (1) \u7531\u5df2\u77e5\u5f97  b \\cos A+a \\cos B=\\frac{2 \\sqrt{3}}{3} b \\sin C .\n\u6240\u4ee5~~~~~~~~~~~ \\\\ \\sin B=\\frac{\\sqrt{3}}{2} . \u56e0\u4e3a  B  \u662f\u9510\u89d2, \u6240\u4ee5  B=\\frac{\\pi}{3} .\n(2) \u7531\u6b63\u5f26\u5b9a\u7406, \u5f97  \\frac{a}{\\sin A}=\\frac{c}{\\sin C}=\\frac{b}{\\sin B}=4 ,\n\n\\\\\u5219  a=4 \\sin A, c=4 \\sin C .\n\u6240\u4ee5  a+c=4 \\sin A+4 \\sin C=4 \\sin A+4 \\sin \\left(\\frac{2 \\pi}{3}-A\\right)~~~~~~\\\\=4 \\sqrt{3} \\sin \\left(A+\\frac{\\pi}{6}\\right) .\n\u7531\u9510\u89d2\u4e09\u89d2\u5f62  A B C  \u53ef\u77e5  \\left\\{\\begin{matrix}\n0 &lt; A &lt; \\displaystyle\\frac{\\pi}{2}&amp; \\\\\n0 &lt;  \\displaystyle\\frac{2 \\pi}{3}-A &lt;  \\displaystyle\\frac{\\pi}{2}  &amp;\n\\end{matrix}\\right. , \u5f97 ~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\ \\frac{\\pi}{6}&lt; A&lt;\\frac{\\pi}{2} , \n\\therefore   \\frac{\\pi}{3} &lt; A+\\frac{\\pi}{6}&lt;\\frac{2 \\pi}{3} , \\therefore   \\frac{\\sqrt{3}}{2} &lt; \\sin \\left(A+\\frac{\\pi}{6}\\right) \\leqslant 1 ,\n\\therefore  6&lt; a+c \\leqslant 4 \\sqrt{3} . ~~~~~~~~~\\\\\\therefore   a+c  \u7684\u53d6\u503c\u8303\u56f4\u4e3a  (6,4 \\sqrt{3}] .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5df2\u77e5  \\triangle A B C  \u4e3a\u9510\u89d2\u4e09\u89d2\u5f62, \u89d2  A, B, C  \u5bf9\u5e94\u7684\u8fb9\u5206\u522b\u4e3a  a, b, c , \u4e14  \\sqrt{3} a \\sin B-b \\cos A=b .\\\\\n(1) \u6c42  A  \u7684\u503c;\n(2) \u82e5  a=2 , \u6c42  2 b-c  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-sqrt-3-sin-a-cos-a-1-rightarrow-sin-left-a-frac-pi-6-right-frac-1-2-therefore-a-frac-pi-3-pi-therefore-a-frac-pi-3-2-frac-a-sin-a-frac-b-sin-b-frac-c-sin-c-frac-4-sqrt-3-therefore-2-b-c-2-times-frac-4-sqrt-3-sin-b-frac-4-sqrt-3-sin-c-frac-4-sqrt-3-left-2-sin-b-sin-left-frac-2-pi-3-b-right-right-frac-4-sqrt-3-cdot-left-frac-3-2-sin-b-frac-sqrt-3-2-cos-b-right-4-sin-left-b-frac-pi-6-right-because-triangle-a-b-c-therefore-left-begin-matrix-0-lt-b-lt-displaystyle-frac-pi-2-amp-0-lt-displaystyle-frac-2-pi-3-b-lt-displaystyle-frac-pi-2-amp-end-matrix-right-frac-pi-6-lt-b-lt-frac-pi-2-0-lt-b-frac-pi-6-lt-frac-pi-3-0-lt-sin-left-b-frac-pi-6-right-lt-frac-sqrt-3-2-0-lt-4-sin-left-b-frac-pi-6-right-lt-2-sqrt-3\"><pre>(1)\u7531\u5df2\u77e5\u5f97    \\sqrt{3} \\sin A-\\cos A=1 , \\Rightarrow  \\sin \\left(A-\\frac{\\pi}{6}\\right)=\\frac{1}{2} ,\\therefore  A=\\frac{\\pi}{3} \u6216\\pi(\u820d),\\therefore A=\\frac{\\pi}{3} \uff1b\n\\\\(2)\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}=\\frac{4}{\\sqrt{3}} ,\n\n\\therefore  2 b-c=2 \\times \\frac{4}{\\sqrt{3}} \\sin B-\\frac{4}{\\sqrt{3}} \\sin C=~~~~~~~~~~~~~~~~~~~~~~~\\\\\\frac{4}{\\sqrt{3}}\\left[2 \\sin B-\\sin \\left(\\frac{2 \\pi}{3}-B\\right)\\right]  =\\frac{4}{\\sqrt{3}} \\cdot\\left(\\frac{3}{2} \\sin B-\\frac{\\sqrt{3}}{2} \\cos B\\right)=4 \\sin \\left(B-\\frac{\\pi}{6}\\right) .~~~~~~~~~\\\\\\because \\triangle A B C  \u4e3a\u9510\u89d2\u4e09\u89d2\u5f62,\\therefore\n\\left\\{\\begin{matrix}\n 0 &lt; B &lt; \\displaystyle\\frac{\\pi}{2} &amp; \\\\\n 0 &lt; \\displaystyle\\frac{2 \\pi}{3}-B &lt; \\displaystyle\\frac{\\pi}{2} &amp;\n\\end{matrix}\\right.\u5f97 , \\frac{\\pi}{6} &lt; B &lt; \\frac{\\pi}{2} , \u53ef\u5f97  0 &lt; B-\\frac{\\pi}{6} &lt; \\frac{\\pi}{3} ,\\\\ \u6240\u4ee5  0&lt;\\sin \\left(B-\\frac{\\pi}{6}\\right)&lt;\\frac{\\sqrt{3}}{2} , \u5219  0 &lt; 4 \\sin \\left(B-\\frac{\\pi}{6}\\right) &lt; 2 \\sqrt{3} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\u5df2\u77e5  \\triangle A B C  \u4e3a\u9510\u89d2\u4e09\u89d2\u5f62,  a, b, c  \u4e3a  \\triangle A B C  \u7684\u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9,  b=2 , \u4e14  A=\\frac{\\pi}{3} , ~~~~~~\\\\\u6c42  \\triangle A B C  \u9762\u79ef\u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-left-frac-sqrt-3-2-2-sqrt-3-right-frac-c-sin-c-frac-2-sin-b-c-frac-2-sin-c-sin-b-rightarrow-s-triangle-a-b-c-frac-1-2-bc-sin-a-frac-sqrt-3-sin-c-sin-b-frac-sqrt-3-sin-left-displaystyle-frac-2-pi-3-b-right-sin-b-frac-sqrt-3-2-frac-3-2-tan-b-left-begin-matrix-0-lt-b-lt-displaystyle-frac-pi-2-amp-0-lt-displaystyle-frac-2-pi-3-b-lt-displaystyle-frac-pi-2-amp-end-matrix-right-frac-pi-6-lt-b-lt-frac-pi-2-tan-b-gt-frac-sqrt-3-3-therefore-frac-sqrt-3-2-lt-frac-sqrt-3-2-frac-3-2-tan-b-lt-2-sqrt-3-therefore-triangle-a-b-c-left-frac-sqrt-3-2-2-sqrt-3-right\"><pre>\u7b54\u6848: \\left(\\frac{\\sqrt{3}}{2}, 2 \\sqrt{3}\\right) ;\u63d0\u793a:\u7531\u6b63\u5f26\u5b9a\u7406  \\frac{c}{\\sin C}=\\frac{2}{\\sin B}, \u5f97  c=\\frac{2 \\sin C}{\\sin B} ,\\Rightarrow  S_{\\triangle A B C}~~~~~~~~~~~~\\\\=\\frac{1}{2} bc\\sin A=\\frac{\\sqrt{3} \\sin C}{\\sin B}=\\frac{\\sqrt{3} \\sin \\left(\\displaystyle\\frac{2\\pi}{3}-B\\right)}{\\sin B}=\\frac{\\sqrt{3}}{2}+\\frac{3}{2 \\tan B} ,\u7531\u9510\u89d2\u4e09\u89d2\u5f62\u53ef\u77e5,~~~~~\\\\\n\\left\\{\\begin{matrix}\n 0 &lt; B &lt; \\displaystyle\\frac{\\pi}{2} &amp; \\\\\n 0 &lt; \\displaystyle\\frac{2 \\pi}{3}-B &lt; \\displaystyle\\frac{\\pi}{2} &amp;\n\\end{matrix}\\right.\u5f97 , \\frac{\\pi}{6} &lt; B &lt; \\frac{\\pi}{2} , \u5219  \\tan B&gt;\\frac{\\sqrt{3}}{3} , \\therefore  \\frac{\\sqrt{3}}{2}&lt;\\frac{\\sqrt{3}}{2}+\\frac{3}{2 \\tan B}&lt;2 \\sqrt{3} ,\\\\\n\\therefore  \\triangle A B C  \u9762\u79ef\u7684\u53d6\u503c\u8303\u56f4\u4e3a  \\left(\\frac{\\sqrt{3}}{2}, 2 \\sqrt{3}\\right) .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\n\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u8bb0  \\triangle A B C  \u7684\u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9\u5206\u522b\u4e3a  a, b, c ,\u5df2\u77e5  \\frac{\\cos A}{1+\\sin A}=\\frac{\\sin 2 B}{1+\\cos 2 B} .~~~~~~~~~~~~~~~~~~\\\\\n(1) \u82e5  C=\\frac{2 \\pi}{3} , \u6c42  B ;\n(2)\u6c42  \\frac{a^{2}+b^{2}}{c^{2}}  \u7684\u6700\u5c0f\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d\"><pre>\u7b54\u6848:(1)\\frac{\\pi}{6},(2)4\\sqrt{2}-5;\u63d0\u793a:(1) \\frac{\\cos A}{1+\\sin A}=\\frac{2 \\sin B \\cos B}{2 \\cos ^{2} B} , \\Rightarrow \\frac{\\cos A}{1+\\sin A}=\\frac{\\sin B}{\\cos B} , \\Rightarrow\\\\\\cos (A+B)=\\sin B  \uff0c  A+B+B=\\frac{\\pi}{2}, \u7531C=\\frac{2 \\pi}{3}\\Rightarrow B=\\frac{\\pi}{6} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u7531(1)\u77e5,  \\left\\{\\begin{array}{l}A+2 B=\\displaystyle\\frac{\\pi}{2} \\\\ A+B+C=\\pi\\end{array} \\Rightarrow\\left\\{\\begin{array}{l}A=C-\\displaystyle\\frac{\\pi}{2} \\\\ B=\\displaystyle\\frac{3 \\pi}{2}-2 c\\end{array}\\right.\\right. ,\u539f\u5f0f=\n\n\\frac{\\cos ^{2} C+\\cos ^{2} 2 C}{\\sin ^{2} C}~~~~~~~~~~~\\\\=\\frac{\\cos ^{2} C+(1-\\sin 2 C)^{2}}{\\sin ^{2} C}=\\frac{2}{\\sin ^{2} C}+4 \\sin ^{2} C-5 \\geq 4 \\sqrt{2}-5,\n\n\n\u5f53\u4e14\u4ec5\u5f53  \\sin C=\\frac{\\sqrt{2}}{2}  \u65f6~\\\\\u53d6\u7b49\u53f7\u3002~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n<\/details>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-111111111111111\">\u4e09\u89d2\u5f62\u7684\u9ad8\u7ebf\u3001\u4e2d\u7ebf\u3001\u89d2\u5e73\u5206\u7ebf<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1. \u5728  \\triangle A B C  \u4e2d,  A B=2, A C=1, \\angle B A C=120^{\\circ}, A H  \u4e3a  \\triangle A B C  \u7684\u9ad8\u7ebf, \u6c42  \\overrightarrow{A B} \\cdot \\overrightarrow{A H}.~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-be35ad2\" data-block-id=\"be35ad2\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-be35ad2-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-035619e\" data-v=\"4\" data-block-id=\"035619e\"><style>@media screen and (min-width:690px){.stk-035619e {flex:var(--stk-flex-grow, 1) 1 calc(66.666\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-c-b-c-sqrt-7-rightarrow-a-h-frac-sqrt-21-7-overrightarrow-a-b-cdot-overrightarrow-a-h-overrightarrow-a-h-overrightarrow-h-b-cdot-overrightarrow-a-h-overrightarrow-a-h-2-frac-3-7\"><pre>\u7b54\u6848 :\\frac{3}{7};\n\u63d0\u793a:\u7531\u4f59\u5f26\u5b9a\u7406\u5f97  B C=\\sqrt{7} , \u7b49\u9762\u79ef\u6cd5~~~~~~\\\\\\Rightarrow  A H=\\frac{\\sqrt{21}}{7}, \\overrightarrow{A B} \\cdot \\overrightarrow{A H}=(\\overrightarrow{A H}+\\overrightarrow{H B}) \\cdot \\overrightarrow{A H}   =~~~~~~~~\\\\|\\overrightarrow{A H}|^{2}=\\frac{3}{7} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4d7e76f\" data-v=\"4\" data-block-id=\"4d7e76f\"><style>@media screen and (min-width:690px){.stk-4d7e76f {flex:var(--stk-flex-grow, 1) 1 calc(33.33400000000002\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/10\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-10-29-170057.jpg\" alt=\"\" class=\"wp-image-3242\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u5df2\u77e5  \\triangle A B C  \u7684\u5185\u89d2  A, B, C  \u6240\u5bf9\u7684\u8fb9\u5206\u522b\u4e3a  a, b, c, C=\\frac{\\pi}{3} , \u82e5  A B  \u8fb9\u4e0a\u7684\u9ad8\u7ebf\u957f\u4e3a  2 \\sqrt{3} , \\\\\u6c42  \\triangle A B C  \u9762\u79ef\u7684\u6700\u5c0f\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-4-sqrt-3-frac-1-2-c-cdot-2-sqrt-3-frac-1-2-a-b-sin-c-rightarrow-a-b-4-c-rightarrow-c-2-a-2-b-2-a-b-rightarrow-a-b-geqslant-16-a-b-4-triangle-a-b-c-4-sqrt-3\"><pre>\u7b54\u6848:4 \\sqrt{3};\u63d0\u793a:\n\u7531\u7b49\u9762\u79ef\u6cd5\u5f97  \\frac{1}{2} c \\cdot 2 \\sqrt{3}=\\frac{1}{2} a b \\sin C, \\Rightarrow a b=4 c , \u7531\u4f59\u5f26\u5b9a\u7406  \\Rightarrow~~~~~~~~~ \\\\c^{2}=a^{2}+b^{2}-a b  \\Rightarrow a b \\geqslant 16 , \u5f53\u4e14\u4ec5\u5f53a=b=4  \u53d6\u7b49\u53f7,   \\triangle A B C  \u9762\u79ef\u7684\u6700\u5c0f\u503c\u4e3a  4 \\sqrt{3} .~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5728  \\triangle A B C  \u4e2d, \u5df2\u77e5  A B=4, A C=7, B C  \u8fb9\u7684\u4e2d\u7ebf  A D=\\frac{7}{2} , \u6c42  B C\u7684\u957f.~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-e608254\" data-block-id=\"e608254\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-e608254-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-bc467bd\" data-v=\"4\" data-block-id=\"bc467bd\"><style>@media screen and (min-width:690px){.stk-bc467bd {flex:var(--stk-flex-grow, 1) 1 66.7\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-9-overrightarrow-a-d-frac-1-2-overrightarrow-a-b-overrightarrow-a-c-rightarrow-cos-a-frac-2-7-rightarrow-b-c-9\"><pre>\u7b54\u6848: 9;\u63d0\u793a:\\overrightarrow{A D}=\\frac{1}{2}(\\overrightarrow{A B}+\\overrightarrow{A C}) , \u4e24\u8fb9\u5e73\u65b9  ~~~~~~~~~~~~~\\\\\\Rightarrow \\cos A=-\\frac{2}{7}, \\Rightarrow B C=9 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3a52a8f\" data-v=\"4\" data-block-id=\"3a52a8f\"><style>@media screen and (min-width:690px){.stk-3a52a8f {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/10\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-10-29-172256.jpg\" alt=\"\" class=\"wp-image-3245\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\u5728  \\triangle A B C  \u4e2d, \u5df2\u77e5  A=60^{\\circ}, B C=2, D  \u4e3a  B C  \u7684\u4e2d\u70b9,\u6c42\u7ebf\u6bb5  A D  \u957f\u5ea6\u7684\u6700\u5927\u503c.~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-sqrt-3-4-b-2-c-2-b-c-quad-rightarrow-b-c-leq-4-overrightarrow-a-d-frac-1-2-overrightarrow-a-b-overrightarrow-a-c-rightarrow-overrightarrow-a-d-2-frac-1-4-left-b-2-c-2-b-c-right-frac-1-4-4-2-b-c-leq-3\"><pre>\u7b54\u6848:\\sqrt{3};\u63d0\u793a:\u7531\u4f59\u5f26\u5b9a\u7406\u5f97  4=b^{2}+c^{2}-b c, \\quad \\Rightarrow b c \\leq 4 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n \\overrightarrow{A D}=\\frac{1}{2}(\\overrightarrow{A B}+\\overrightarrow{A C}), \\Rightarrow \\overrightarrow{A D}^{2}=\\frac{1}{4}\\left(b^{2}+c^{2}+b c\\right)=\\frac{1}{4}(4+2 b c) \\leq 3 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u5728  \\triangle A B C  \u4e2d,  A B=6, A C=2, A D  \u4e3a  \\angle B A C  \u7684\u89d2\u5e73\u5206\u7ebf,  A D   =\\sqrt{3} , \u6c42\\triangle A B C  \u7684\u9762\u79ef.\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre><\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-8bdf931\" data-block-id=\"8bdf931\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-8bdf931-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-e9c5474\" data-v=\"4\" data-block-id=\"e9c5474\"><style>@media screen and (min-width:690px){.stk-e9c5474 {flex:var(--stk-flex-grow, 1) 1 calc(72\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-4-sqrt-2-s-triangle-a-b-c-s-triangle-a-b-d-s-triangle-a-c-d-frac-1-2-a-b-cdot-a-c-cdot-sin-2-theta-frac-1-2-a-b-cdot-a-d-cdot-sin-theta-frac-1-2-a-d-cdot-a-c-cdot-sin-theta-rightarrow-cos-theta-frac-sqrt-3-3-sin-theta-frac-sqrt-6-3-sin-2-theta-frac-2-sqrt-2-3-rightarrow-s-triangle-a-b-c-4-sqrt-2\"><pre>\u7b54\u68484 \\sqrt{2}:\u63d0\u793a: S_{\\triangle A B C}=S_{\\triangle A B D}+S_{\\triangle A C D},~~~~~~~~~~~~~~~~~~~~~\\\\\\frac{1}{2} A B \\cdot A C \\cdot \\sin 2 \\theta=\\frac{1}{2} A B \\cdot A D \\cdot \\sin \\theta+\\frac{1}{2} A D \\cdot A C \\cdot \\sin \\theta, \\\\\\Rightarrow \\cos \\theta=\\frac{\\sqrt{3}}{3}, \\sin \\theta=\\frac{\\sqrt{6}}{3}, \\sin 2 \\theta=\\frac{2 \\sqrt{2}}{3} \\Rightarrow S_{\\triangle A B C}=4 \\sqrt{2} .<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5c2a4c7\" data-v=\"4\" data-block-id=\"5c2a4c7\"><style>@media screen and (min-width:690px){.stk-5c2a4c7 {flex:var(--stk-flex-grow, 1) 1 calc(28\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/10\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-10-29-145348.jpg\" alt=\"\" class=\"wp-image-3251\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\\triangle A B C  \u7684\u5185\u89d2  A, B, C  \u7684\u5bf9\u8fb9\u5206\u522b\u4e3a  a, b, c , \u5df2\u77e5  2 a+b=2 c \\cos B , \u82e5  C D \u662f\u89d2  C  \u7684\u5e73~~~\\\\\u5206\u7ebf,  A D=2 \\sqrt{7}, D B=\\sqrt{7} , \u5219  C D  \u7684\u957f\u4e3a (\\quad )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA. 3\\quad \nB. 2\\quad \nC.  2 \\sqrt{2} \\quad \nD.  3 \\sqrt{2} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre><\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-c0fb702\" data-block-id=\"c0fb702\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c0fb702-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-675c87f\" data-v=\"4\" data-block-id=\"675c87f\"><style>@media screen and (min-width:690px){.stk-675c87f {flex:var(--stk-flex-grow, 1) 1 calc(70\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-b-rightarrow-c-frac-2-pi-3-frac-a-c-b-c-frac-a-d-b-d-2-b-c-x-a-c-2-x-triangle-a-b-c-a-b-2-a-c-2-b-c-2-2-a-c-cdot-b-c-cdot-cos-angle-a-c-b-rightarrow-x-3-s-triangle-b-c-d-s-triangle-a-c-d-s-triangle-a-b-c-rightarrow-c-d-2\"><pre>\u7b54\u6848:B;\u63d0\u793a:\u7531\u5df2\u77e5  \\Rightarrow C=\\frac{2 \\pi}{3} . \u7531\u89d2\u5e73\u5206\u7ebf\u5b9a\u7406\u77e5 ~~~~\\\\ \\frac{A C}{B C}=\\frac{A D}{B D}=2 ,\n\u8bbe  B C=x , \u5219  A C=2 x , \u5728  \\triangle A B C  \u4e2d, ~~~~~\\\\\u7531\u4f59\u5f26\u5b9a\u7406A B^{2}=A C^{2}+B C^{2}-2 A C \\cdot B C \\cdot \\cos \\angle A C B, \\\\\\Rightarrow x=3,S_{\\triangle B C D}+S_{\\triangle A C D}=S_{\\triangle A B C} \\Rightarrow C D=2 .~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0d3da05\" data-v=\"4\" data-block-id=\"0d3da05\"><style>@media screen and (min-width:690px){.stk-0d3da05 {flex:var(--stk-flex-grow, 1) 1 calc(30\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/10\/jsjx3.png\" alt=\"\" class=\"wp-image-3254\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n \n","protected":false},"excerpt":{"rendered":"<p>\u4e09\u89d2\u51fd\u6570\u7684\u6027\u8d28\uff0c\u56fe\u8c61\u53ca\u53d8\u6362<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-3178","post","type-post","status-publish","format-standard","hentry","category-2"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/3178","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3178"}],"version-history":[{"count":0,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/3178\/revisions"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3178"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}