{"id":17135,"date":"2010-12-06T16:10:11","date_gmt":"2010-12-06T08:10:11","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=17135"},"modified":"2021-10-06T16:30:22","modified_gmt":"2021-10-06T08:30:22","slug":"%e6%ad%a3%e5%bc%a6%e5%ae%9a%e7%90%86law-of-sine","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%ad%a3%e5%bc%a6%e5%ae%9a%e7%90%86law-of-sine\/","title":{"rendered":"\u6b63\u5f26\u5b9a\u7406(Law of sine)"},"content":{"rendered":"

\u6b63\u5f26\u5b9a\u7406(Law of sine)<\/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<\/span><\/strong><\/p>\n

\u6b63\u5f26\u5b9a\u7406\uff1a\u82e5 \\(\\Delta{ABC}\\) \u7684\u4e09\u908a\u9577 \\(\\overline{BC}=a,\\overline{CA}=b,\\overline{AB}=c\\)\uff0c<\/p>\n

\u5247\u6046\u6709\u6027\u8cea \\(\\displaystyle\\frac{a}{\\sin{A}}=\\frac{b}{\\sin{B}}=\\frac{c}{\\sin{C}}=2R\\)\uff0c\u6b64\u7a31\u70ba\u6b63\u5f26\u5b9a\u7406\u3002<\/p>\n

\u8b49\u660e\uff1a\u56e0\u70ba \\(a\\Delta{ABC}=\\frac{1}{2}bc\\sin{A}=\\frac{1}{2}ca\\sin{B}=\\frac{1}{2}ab\\sin{C}\\)\uff0c<\/p>\n

\u540c\u4e58\u4e8c\u500d\u5f97 \\(bc\\sin{A}=ca\\sin{B}=ab\\sin{C}\\)<\/p>\n

\u540c\u9664 \\(abc\\) \u5f97 \\(\\displaystyle\\frac{\\sin{A}}{a}=\\frac{\\sin{B}}{b}=\\frac{\\sin{C}}{c}\\)\uff0c<\/p>\n

\u53d6\u5176\u5012\u6578\u5f97 \\(\\displaystyle\\frac{a}{\\sin{A}}=\\frac{b}{\\sin{B}}=\\frac{c}{\\sin{C}}~-(1)\\)\u3002<\/p>\n

\u5728\u4e0d\u5931\u4e00\u822c\u6027\u7684\u60c5\u5f62\u4e0b\uff0c\u6211\u5011\u4ee5\u5713\u5167\u63a5\u92b3\u89d2\u4e09\u89d2\u5f62\u9032\u884c\u8b49\u660e\uff0c\u5982\u5716\u4e00\u6240\u793a\u3002\u00a0\u00a0<\/p>\n

\"\"<\/p>\n

\\(\\overline{BC}\\) \u4e2d\u9ede \\(M\\)\uff0c\u5247 \\(\\overline{BM}=\\frac{1}{2}\\overline{BC}=\\frac{a}{2}\\)\uff0c\u800c \\(\\angle{BOM}=\\frac{1}{2}\\angle{BOC}=\\frac{1}{2}(2\\angle{A})=\\angle{A}\\)\uff0c<\/p>\n

\u6839\u64da\u6b63\u5f26\u51fd\u6578\u5b9a\u7fa9\uff1a\\(\\displaystyle\\sin{A}=\\frac{\\frac{a}{2}}{R}=\\frac{a}{2R}\\)\uff0c\\(\\therefore\\displaystyle\\frac{a}{\\sin{A}}=2R\\)<\/p>\n

\u4ee3\u5165 \\((1)\\) \u5f97 \\(\\displaystyle\\frac{a}{\\sin{A}}=\\frac{b}{\\sin{B}}=\\frac{c}{\\sin{C}}=2R\\)\uff0c\u5f97\u8b49\u3002<\/p>\n

\u6b63\u5f26\u5b9a\u7406\u7279\u4f8b\uff1a\u5982\u679c\u6211\u5011\u5047\u8a2d\u4e09\u89d2\u5f62\u7684\u5916\u63a5\u5713\u76f4\u5f91\u7684\u9577\u5ea6\u70ba \\(1\\)\uff0c\u5247\u6b63\u5f26\u5b9a\u7406\u6703\u8f49\u63db\u6210 \\(a=\\sin{A},b=\\sin{B},c=\\sin{C}\\)\uff0c\u9019\u500b\u7d50\u8ad6\u986f\u793a\u51fa\u5167\u63a5\u65bc\u76f4\u5f91\u70ba \\(1\\) \u7684\u5713\uff0c\u6b64\u4e09\u89d2\u5f62\u908a\u9577\u7b49\u65bc\u5c0d\u89d2\u7684\u6b63\u5f26\u503c\uff0c\u800c\u9019\u500b\u908a\u9577\u5c31\u662f\u4e09\u89d2\u5f62\u5728\u5713\u4e0a\u6240\u5f35\u958b\u7684\u5f26\u9577\uff0c\u5982\u5716\u4e8c\u6240\u793a\u3002<\/p>\n

\"\"<\/p>\n

\u9ad8\u4e2d\u7684\u6b63\u5f26\u5b9a\u7406\u662f\u570b\u4e2d\u5e7e\u4f55\u4e2d\u300c\u4e09\u89d2\u5f62\u4e2d\uff0c\u5927\u908a\u5c0d\u5927\u89d2\uff0c\u5c0f\u908a\u5c0d\u5c0f\u89d2\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u300d\u7684\u5177\u9ad4\u91cf\u5316\u5b9a\u7406\uff0c\u900f\u904e\u6b63\u5f26\u5b9a\u7406\uff0c\u6211\u5011\u53ef\u4ee5\u5c0e\u51fa\u6b63\u5207\u5b9a\u7406(Law of tangent)\uff1a<\/p>\n

\\(\\displaystyle\\frac{a-b}{a+b}=\\frac{\\tan(\\frac{A-B}{2})}{\\tan(\\frac{A+B}{2})}\\)\u3002<\/p>\n

\u8b49\u660e\uff1a<\/p>\n

\\(\\begin{array}{ll}\\displaystyle\\frac{a-b}{a+b}&\\displaystyle=\\frac{2R\\sin{A}-2R\\sin{B}}{2R\\sin{A}+2R\\sin{B}}=\\frac{\\sin{A}-\\sin{B}}{\\sin{A}+\\sin{B}}\\\\&\\displaystyle=\\frac{2\\sin(\\frac{A-B}{2}\\cos(\\frac{A+B}{2}))}{2\\sin(\\frac{A+B}{2})\\cos(\\frac{A-B}{2})}\\\\&\\displaystyle=\\frac{\\tan(\\frac{A-B}{2})}{\\tan(\\frac{A+B}{2})}\\end{array}\\)<\/p>\n

\u6b63\u5f26\u5b9a\u7406\u63a8\u5ee3\uff1a\u56db\u9762\u9ad4 \\(OABC\\) \u4e2d\uff0c\u5982\u5716\u4e09\u6240\u793a\uff0c\u6211\u5011\u900f\u904e\u6b63\u5f26\u5b9a\u7406\u7684\u61c9\u7528\u53ef\u5f97\u4e0b\u5217\u6027\u8cea\uff1a<\/p>\n

\\(\\sin\\angle{OAB}\\cdot\\sin\\angle{OBC}\\cdot\\sin\\angle{OCA}=\\sin\\angle{OAC}\\cdot\\sin\\angle{OCB}\\cdot\\sin\\angle{OBA}\\)<\/p>\n

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

\u6b63\u5f26\u5b9a\u7406\u9084\u53ef\u4ee5\u5ef6\u4f38\u81f3\u5149\u5b78\u7684\u53f8\u4e43\u8033\u5b9a\u5f8b(Snell’s Law)\uff0c\u9019\u500b\u516c\u5f0f\u662f\u7528\u4f86\u63cf\u8ff0\u5149\u7684\u884c\u9032\u8def\u5f91\uff0c\u5176\u5165\u5c04\u89d2 \\(\\angle{AOP}=\\theta_1\\) \u7684\u6b63\u5f26\u51fd\u6578\u503c \\(\\sin\\theta_1\\)\u3001\u6298\u5c04\u89d2 \\(\\angle{BOQ}=\\theta_2\\) \u7684\u6b63\u5f26\u51fd\u6578\u503c \\(\\sin\\theta_2\\) \u8207\u5149\u5728\u4e0d\u540c\u4ecb\u8cea\u7684\u901f\u7387 \\(v_1,v_2\\) \u7684\u95dc\u4fc2\uff0c\u5982\u5716\u56db\u6240\u793a\uff1a<\/p>\n

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

\u56e0\u70ba\u5149\u7d93\u904e\u5169\u500b\u4ecb\u8cea\u7684\u6642\u9593 \\(\\displaystyle{t}=\\frac{\\overline{PO}}{v_1}+\\frac{\\overline{OQ}}{v_2}=\\frac{\\sqrt{a^2+x^2}}{v_1}+\\frac{\\sqrt{b^2+(d-x)^2}}{v_2}\\)\uff0c<\/p>\n

\u89e3\u51fa \\(\\frac{dt}{dx}=0\\)\uff0c\u5373\u5149\u5728\u5169\u500b\u4ecb\u8cea\u9593\u6240\u8cbb\u6642\u9593\u7684\u6700\u5c0f\u503c\uff0c\u5fae\u5206\u5f97<\/p>\n

\\(\\displaystyle\\frac{1}{v_2}\\cdot\\frac{1}{2}(a^2+x^2)^{-\\frac{1}{2}}\\cdot{2}x+\\frac{1}{v_2}\\cdot\\frac{1}{2}[b^2+(d-x)^2]^{\\frac{1}{2}}\\cdot{2}(d-x)\\cdot(-1)=0\\)\uff0c<\/p>\n

\u6574\u7406\u4e0a\u5f0f\u5f97 \\(\\displaystyle\\frac{x}{v_1\\cdot\\sqrt{a^2+x^2}}=\\frac{d-x}{v_2\\cdot\\sqrt{b^2+(d-x)^2}}\\)\uff0c<\/p>\n

\u5373 \\(\\displaystyle\\frac{\\frac{x}{a^2+x^2}}{v_1}=\\frac{\\frac{d-x}{b^2+(d-x)^2}}{v_2}\\)\uff0c<\/p>\n

\u6839\u64da\u4e09\u89d2\u51fd\u6578\u5b9a\u7fa9\u5f97 \\(\\displaystyle\\frac{\\sin\\theta_1}{v_1}=\\frac{\\sin\\theta_2}{v_2}\\)\uff0c\u900f\u904e\u4e00\u4e9b\u8f49\u63db\uff0c\u5c31\u53ef\u5c07\u4e0a\u5f0f\u5beb\u6210\u53f8\u4e43\u8033\u5b9a\u5f8b\uff1a<\/p>\n

\\(\\displaystyle\\frac{\\sin\\theta_1}{\\sin\\theta_2}=\\frac{v_1}{v_2}=\\frac{n_2}{n_1}\\)\uff0c<\/p>\n

\u5176\u4e2d \\(n_1,n_2\\) \u5206\u5225\u662f\u5169\u500b\u4ecb\u8cea\u7684\u6298\u5c04\u7387(refractive index)\u3002<\/p>\n

\u53f8\u4e43\u8033\u5b9a\u5f8b\u662f\u6e6f\u746a\u65af\uff0e\u54c8\u529b\u7279 (Thomas Harriot, 1560-1621) \u57281602 \u5e74\u6240\u767c\u73fe\u7684\uff0c\u96d6\u7136\u4ed6\u66fe\u8207\u514b\u535c\u52d2 (JohannesKepler, 1571-1630) \u901a\u4fe1\u6642\u8ac7\u8ad6\u6b64\u5b9a\u5f8b\uff0c\u53ef\u60dc\uff0c\u4ed6\u672a\u767c\u8868\u6b64\u7d50\u679c\u3002\u5230\u4e861621 \u5e74\uff0c\u53f8\u4e43\u8033 (Willebrord Snellius, 1580-1626) \u5c0e\u51fa\u6b64\u4e00\u76f8\u95dc\u6578\u5b78\u6046\u7b49\u5f0f\uff0c\u53ef\u662f\u7d42\u5176\u4e00\u751f\u4e5f\u672a\u66fe\u51fa\u7248\u3002<\/p>\n

\u76f4\u52301637 \u5e74\uff0c\u7b1b\u5361\u5152 (Ren\u00e9 Descartes, 1596-1650) \u5728\u5176\u4f5c\u54c1\u300a\u65b9\u6cd5\u8ad6\u300b(Discourse on Method) \u5229\u7528\u6709\u555f\u767c\u6027\u7684\u52d5\u91cf\u5b88\u6046\u539f\u7406\u5c0e\u51fa\u6b64\u4e00\u5b9a\u5f8b\uff0c\u800c\u4e14\uff0c\u4ed6\u9084\u5229\u7528\u6b64\u4e00\u5b9a\u5f8b\u89e3\u6c7a\u8a31\u591a\u5149\u5b78\u65b9\u9762\u7684\u554f\u984c\u3002\u6700\u5f8c\uff0c\u8cbb\u746a (Pierre de Fermat, 1601-1665) \u63a1\u53d6\u5149\u884c\u9032\u7684\u6700\u77ed\u6642\u9593\u539f\u7406\uff0c\u800c\u8b49\u660e\u6b64\u4e00\u5b9a\u5f8b\u7b49\u5f0f\uff0c\u6211\u5011\u4e0a\u4e00\u6bb5\u6240\u8ff0\u7684\u8b49\u660e\u904e\u7a0b\uff0c\u5c31\u662f\u4eff\u81ea\u4ed6\u7684\u7248\u672c\u3002<\/p>\n

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

    \n
  1. \u6bdb\u723e(Eli Maor)\u8457\uff08\u80e1\u5b88\u4ec1\u8b6f\uff09\uff0c\u300a\u6bdb\u8d77\u4f86\u8aaa\u4e09\u89d2\u300b(Trigonometric Delights)\uff0c\u53f0\u5317\uff1a\u5929\u4e0b\u9060\u898b\u51fa\u7248\u793e\uff0c2000\u5e74\u3002<\/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
  3. http:\/\/en.wikipedia.org\/wiki\/Law_of_sines<\/a>\u3002<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u672c\u6587\u5f9e\u6b63\u5f26\u5b9a\u7406\u8ac7\u5230\u53f8\u4e43\u723e\u7684\u6298\u5c04\u5b9a\u5f8b\uff0c\u65e2\u6d89\u53ca\u7279\u4f8b\uff0c\u4e5f\u95dc\u7167\u5230\u5ef6\u4f38\uff0c\u5c0d\u65bc\u76f8\u95dc\u77e5\u8b58\u9032\u884c\u4e86\u6709\u8da3\u7684\u9023\u7d50\uff0c\u662f\u9ad8\u4e2d\u6559\u5e2b\u503c\u5f97\u501f\u93e1\u7684\u8f14\u52a9\u6559\u6750\u7de8\u5beb\u793a\u7bc4\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":[398],"class_list":["post-17135","post","type-post","status-publish","format-standard","hentry","category-math03-03","category-math03","category-mathematics00","tag-398","loop-entry","cat-222","cat-219","cat-111","no-thumbnail"],"views":28282,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/17135","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=17135"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/17135\/revisions"}],"predecessor-version":[{"id":89233,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/17135\/revisions\/89233"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=17135"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=17135"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=17135"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}