{"id":16923,"date":"2010-12-03T17:43:36","date_gmt":"2010-12-03T09:43:36","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=16923"},"modified":"2021-10-06T16:30:23","modified_gmt":"2021-10-06T08:30:23","slug":"%e9%a4%98%e5%bc%a6%e5%ae%9a%e7%90%86law-of-cosine","status":"publish","type":"post","link":"http:\/\/localhost\/%e9%a4%98%e5%bc%a6%e5%ae%9a%e7%90%86law-of-cosine\/","title":{"rendered":"\u9918\u5f26\u5b9a\u7406(Law of cosine)"},"content":{"rendered":"

\u9918\u5f26\u5b9a\u7406(Law of cosine)<\/span><\/strong>
\n\u570b\u7acb\u862d\u967d\u5973\u4e2d\u6578\u5b78\u79d1\u9673\u654f\u6667\u8001\u5e2b\/\u570b\u7acb\u81fa\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u6d2a\u842c\u751f\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/strong><\/span><\/p>\n

\u9918\u5f26\u5b9a\u7406\uff1a\u82e5 \\(\\Delta{ABC}\\) \u7684\u4e09\u908a\u9577 \\(\\overline{BC}=a,\\overline{CA}=b,\\overline{AB}=c\\)\uff0c\u5247\u6046\u6709\u6027\u8cea<\/p>\n

\\(a^2=b^2+c^2-2bc\\cos{A}\\)
\n\\(b^2=a^2+c^2-2ac\\cos{B}\\)\uff0c\u6b64\u7a31\u9918\u5f26\u5b9a\u7406\u3002
\n\\(c^2=a^2+b^2-2ab\\cos{C}\\)<\/p>\n

\u8b49\u660e\uff1a\u5728\u4e0d\u5931\u4e00\u822c\u6027\u7684\u60c5\u5f62\u4e0b\uff0c\u6211\u5011\u4ee5\u92b3\u89d2\u4e09\u89d2\u5f62 \\(ABC\\) \u9032\u884c\u8b49\u660e\uff0c\u5982\u5716\u4e00\u6240\u793a\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u904e \\(C\\) \u9ede\u4f5c \\(\\overline{AB}\\) \u4e4b\u5782\u7dda\u4ea4 \\(\\overline{AB}\\) \u65bc \\(D\\) \u9ede\uff0c
\n\u6839\u64da\u4e09\u89d2\u51fd\u6578\u7684\u5b9a\u7fa9\u5f97 \\(\\overline{AD}=b\\cos{A}\\)\uff0c\u4e14 \\(\\overline{CD}=b\\sin{A}\\)\uff0c
\n\u53c8 \\(\\overline{BD}=\\overline{AB}-\\overline{AD}=c-b\\cos{A}\\)
\n\u5229\u7528\u7562\u6c0f\u5b9a\u7406\u5f97\uff1a\\(a^2={\\overline{BC}}^2={\\overline{CD}}^2+{\\overline{BD}}^2=(b\\sin{A})^2+(c-b\\cos{A})^2=b^2+c^2-2bc\\cos{A}\\)\uff0c\u4e0a\u8ff0\u8b49\u660e\u65b9\u5f0f\u662f\u4e00\u822c\u9ad8\u4e2d\u6578\u5b78\u6559\u79d1\u66f8\u5e38\u7528\u7684\u65b9\u6cd5\uff0c\u5176\u9918\u5169\u5f0f\uff0c\u540c\u7406\u53ef\u8b49\u3002<\/p>\n

\u9918\u5f26\u5b9a\u7406\u7279\u4f8b\uff1a\u5982\u679c\u6211\u5011\u5047\u8a2d\u4e09\u89d2\u5f62\u70ba\u76f4\u89d2\u4e09\u89d2\u5f62\uff0c\u5373 \\(\\angle{A}=90^\\circ\\)\uff0c\u5247\u9918\u5f26\u5b9a\u7406\u6703\u8f49\u63db\u6210\\(a^2=b^2+c^2-2bc\\cos{90}^\\circ=b^2+c^2\\)\uff0c\u9019\u500b\u7d50\u8ad6\u5373\u7562\u6c0f\u5b9a\u7406\u3002<\/p>\n

Timothy A. Sipka \u7684\u4e00\u500b\u8b49\u6cd5\uff08\u6536\u5165Proof without Words\uff09\uff0c\u4e5f\u5f9e\u300a\u5e7e\u4f55\u539f\u672c\u300b\u7684\u7562\u6c0f\u5b9a\u7406\u9762\u7a4d\u5716\u5f62\u51fa\u767c\uff0c\u53bb\u8b49\u660e\u9918\u5f26\u5b9a\u7406\uff0c\u5982\u5716\u4e8c\u6240\u793a\uff0c\u6700\u5f8c\u5229\u7528\u7562\u6c0f\u5b9a\u7406 \\(\\overline{AC}^2=\\overline{AD}^2+\\overline{CD}^2\\)\uff0c\u5373\u5f97\u9918\u5f26\u5b9a\u7406\uff1a<\/p>\n

\\(c^2=(b\\sin\\theta)^2+(a-b\\sin\\theta)^2=a^2+b^2-2ab\\cos\\theta\\)\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u5728 Proof without Words \u4e2d\u5171\u6709\u4e09\u500b\u8b49\u660e\u9918\u5f26\u5b9a\u7406\u7684\u65b9\u5f0f\uff0c<\/p>\n

\u7b2c\u4e8c\u500b\u8b49\u660e\u65b9\u6cd5\u662fSidey H. Kung\u5229\u7528\u5713\u5167\u51aa\u6027\u8cea\uff1a\u53d6\u5713\u5167\u63a5\u76f4\u89d2\u4e09\u89d2\u5f62 \\(ABC\\)\uff0c\u5982\u5716\u4e09\u6240\u793a\uff0c<\/p>\n

\u5176\u659c\u908a \\(\\overline{AC}=2a\\)\uff0c\u4ee4 \\(\\angle{ABC}=\\theta\\)\uff0c\u5247 \\(\\overline{BC}=2a\\cos\\theta\\)\uff0c<\/p>\n

\\(\\overline{EG}\\)\u00a0\u5176\u4e0a\u7dda\u6bb5 \\(\\overline{OF}=C\\)\uff0c\u5247 \\(\\overline{FG}=a-c\\)\uff0c\u4ee4 \\(\\overline{CF}=b\\)\uff0c\u5f97 \\(\\overline{BF}=2a\\cos\\theta-b\\)\uff0c<\/p>\n

\u6700\u5f8c\u6839\u64da\u5713\u5167\u51aa\u6027\u8cea \\(\\overline{FB}\\times\\overline{FC}=\\overline{FE}\\times\\overline{FG}\\)\uff0c\u4ee3\u5165\u4e0a\u8ff0\u5047\u8a2d\u689d\u4ef6\uff0c<\/p>\n

\u5373 \\((2a\\cos\\theta-b)b=(a-c)(a+c)\\)\uff0c\u6574\u7406\u5f97\u9918\u5f26\u5b9a\u7406 \\(c^2=a^2+b^2-2ab\\cos\\theta\\)\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u5716\u4e09\uff1a\u7531\u5713\u5167\u51aa\u6027\u8cea\u8b49\u660e\u9918\u5f26\u5b9a\u7406\u3002<\/strong><\/span><\/p>\n

\u7b2c\u4e09\u500b\u8b49\u660e\u65b9\u6cd5\u662fSidney H. Kung\u5229\u7528\u6258\u52e4\u5bc6\u5b9a\u7406\uff08Ptolemy\u2019s Theorem\uff09\uff1a<\/p>\n

\u82e5 \\(ABCD\\)\u00a0\u70ba\u5713\u5167\u63a5\u7b49\u8170\u68af\u5f62\uff0c\u5982\u5716\u56db\u6240\u793a\uff0c\u5176\u4e2d\u7dda\u6bb5 \\(AB\\) \u5e73\u884c\u65bc\u7dda\u6bb5 \\(CD\\)\uff0c<\/p>\n

\u4ee4 \\(\\overline{AB}=a\\)\uff0c\\(\\overline{AD}=\\overline{BC}=b\\)\uff0c\u4e14 \\(\\angle{DAB}=\\angle{ABC}=\\theta\\)\uff0c\u5c0d\u89d2\u7dda\u9577 \\(\\overline{AC}=\\overline{BD}=c\\)\uff0c<\/p>\n

\u5229\u7528\u9918\u5f26\u51fd\u6578\u5b9a\u7fa9\u5f97 \\(\\overline{CD}=a+2b\\cos(\\pi-\\theta)\\)\uff0c<\/p>\n

\u6839\u64da\u6258\u52d2\u5bc6\u5b9a\u7406\uff1a\u5713\u5167\u63a5\u56db\u908a\u5f62\uff0c\u5176\u5c0d\u89d2\u7dda\u76f8\u4e58\u503c\u7b49\u65bc\u5c0d\u908a\u9577\u76f8\u4e58\u4e4b\u548c\uff0c<\/p>\n

\u5373 \\(\\overline{AC}\\times\\overline{BD}=\\overline{AB}\\times\\overline{CD}+\\overline{AD}\\times\\overline{BC}\\)\uff0c<\/p>\n

\u4ee3\u5165\u5047\u8a2d\u689d\u4ef6\u5373 \\(c\\times{c}=a\\times[a+2b\\cos(\\pi-\\theta)]+b\\times{b}\\)\uff0c<\/p>\n

\u6574\u7406\u5f97\u9918\u5f26\u5b9a\u7406 \\(c^2=a^2+b^2-2ab\\cos\\theta\\)\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u9664\u4e86Proof without Words\u4e2d\u4e09\u500b\u8b49\u660e\u9918\u5f26\u5b9a\u7406\u7684\u65b9\u5f0f\u5916\uff0c\u9ad8\u4e2d\u6578\u5b78\u8001\u5e2b\u5728\u6559\u5b78\u4e0a\u4e5f\u5e38\u5f15\u9032\u5e73\u9762\u5169\u9ede\u8ddd\u96e2\u516c\u5f0f\u7684\u8b49\u660e\u8207\u8907\u6578\u7684\u8b49\u660e\u3002<\/p>\n

\u5e73\u9762\u5169\u9ede\u8ddd\u96e2\u516c\u5f0f\u7684\u8b49\u660e\uff1a\u7d66\u5b9a\u4efb\u610f\u4e09\u89d2\u5f62 \\(ABC\\) \u8207\u5e73\u9762\u76f4\u89d2\u5750\u6a19\u7cfb\uff0c\u5982\u5716\u4e94\u6240\u793a\uff0c<\/p>\n

\u4ee4 \\(C\\) \u9ede\u70ba\u539f\u9ede \\((0,0)\\)\uff0c\u4e14\u4e09\u908a\u9577 \\(\\overline{BC}=a,\\overline{CA}=b,\\overline{AB}=c\\)\uff0c<\/p>\n

\u5f97 \\(B\\) \u9ede\u5750\u6a19\u70ba \\((a,0)\\)\u3001\\(A\\) \u9ede\u5750\u6a19\u70ba \\((b\\cos{C},b\\sin{C})\\)\uff0c\u5229\u7528\u5169\u9ede\u4e4b\u9593\u7684\u7dda\u6bb5\u516c\u5f0f\uff1a<\/p>\n

\\(c^2=\\overline{AB}^2=(b\\cos{C}-a)^2+(b\\sin{C}-0)^2=a^2+b^2-2ab\\cos{C}\\)\uff0c<\/p>\n

\u6b64\u5373\u70ba\u9918\u5f26\u5b9a\u7406\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u8907\u6578\u8ddd\u96e2\u516c\u5f0f\u7684\u8b49\u660e\uff1a\u7d66\u5b9a\u4efb\u610f\u4e09\u89d2\u5f62 \\(ABC\\) \u8207\u8907\u6578\u5e73\u9762\u5750\u6a19\u7cfb\uff0c\u5982\u5716\u516d\u6240\u793a\uff0c<\/p>\n

\u4ee4 \\(C\\) \u70ba\u539f\u9ede \\((0,0)\\)\uff0c\u4e14\u4e09\u908a\u9577 \\(\\overline{BC}=a,\\overline{CA}=b,\\overline{AB}=c\\)\uff0c<\/p>\n

\u5176\u4e2d \\(A\\) \u9ede\u70ba \\(z_1=b(\\cos\\theta_1+i\\sin\\theta_1)\\)\uff0c\\(B\\) \u9ede\u70ba\\(z_2=a(\\cos\\theta_2+i\\sin\\theta_2)\\)\uff0c<\/p>\n

\u5f97 \\(A\\) \u9ede\u5750\u6a19\u70ba \\((b\\cos\\theta_1,b\\sin\\theta_1)\\)\u3001\\(B\\) \u9ede\u5750\u6a19\u70ba \\((a\\cos\\theta_2,a\\sin\\theta_2)\\)\uff0c<\/p>\n

\u65bc\u662f\u5169\u8907\u6578\u5dee\u70ba \\(z_1-z_2=(b\\cos\\theta_1-a\\cos\\theta_2)+i(b\\sin\\theta_1-a\\sin\\theta_2)\\)\uff0c<\/p>\n

\u5229\u7528\u8907\u6578\u5e73\u9762\u5169\u9ede\u7dda\u6bb5\u516c\u5f0f\uff1a<\/p>\n

\\(\\begin{array}{ll}c^2=\\overline{AB}^2&=|z_1-z_2|^2=(z_1-z_2)\\cdot(\\overline{z_1-z_2})\\\\&=(b\\cos\\theta_1-a\\cos\\theta_2)^2+(b\\sin\\theta_1-a\\sin\\theta_2)^2\\\\&=a^2+b^2-2ab(cos\\theta_1\\cos\\theta_2+\\sin\\theta_1\\sin\\theta_2)\\\\&=a^2+b^2-2ab\\cos(\\theta_1-\\theta_2)\\\\&=a^2+b^2-2bc\\cos{C}\\end{array}\\)\uff0c<\/p>\n

\u6b64\u5373\u70ba\u9918\u5f26\u5b9a\u7406\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u53c3\u8003\u8cc7\u6599\uff1a<\/strong><\/p>\n

    \n
  1. Roger B. Nelsen(1993).Proof without Words: Exercises in Visual Thinking, Washington D.C.\uff1aThe Mathematical Association of America.<\/li>\n
  2. \u8521\u8070\u660e\uff0c\u300a\u6578\u5b78\u7684\u767c\u73fe\u8da3\u8ac7\u300b\uff0c\u53f0\u5317\uff1a\u4e09\u6c11\u66f8\u5c40\uff0c2000\u5e74\u3002<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u672c\u6587\u8aaa\u660e\u9918\u5f26\u5b9a\u5f8b\u7684\u5e7e\u500b\u8b49\u660e\u53ca\u5176\u5ef6\u4f38\uff0c\u5c24\u5176\u4ecb\u7d39\u5716\u8aaa\u4e00\u9ad4\uff08proof without words\uff09\u4e4b\u8b49\u6cd5\uff0c\u503c\u5f97\u53c3\u8003\u3002<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[222,219,111],"tags":[387,503],"class_list":["post-16923","post","type-post","status-publish","format-standard","hentry","category-math03-03","category-math03","category-mathematics00","tag-387","tag-503","avhec_catgroup-all","loop-entry","cat-222","cat-219","cat-111","no-thumbnail"],"views":52898,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/16923","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=16923"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/16923\/revisions"}],"predecessor-version":[{"id":89250,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/16923\/revisions\/89250"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=16923"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=16923"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=16923"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}