三角函数图像曲线问题，诱导公式推余弦曲线？ 求所有三角函数的性质公式和图像
\u4f59\u5f26\u66f2\u7ebf\u7684\u6027\u8d28\u9ad8\u4e2d\u65b0\u6559\u6751\u300a\u6570\u5b66\u300b\u7b2c\u4e00\u518c\uff08\u4e0b\uff09\u00a74.8 \u6b63\u5f26\u51fd\u6570\u3001\u4f59\u5f26\u51fd\u6570\u7684\u56fe\u8c61\u548c\u6027\u8d28\uff08\u4e00\uff09\u6b63\u5f26\u51fd\u6570\u3001\u4f59\u5f26\u51fd\u6570\u7684\u56fe\u8c61 \u5355\u4f4d\uff1a\u6cb3\u5357\u7701\u6d4e\u6e90\u5e02\u7b2c\u4e00\u4e2d\u5b66 \u4f5c\u8005\uff1a\u77f3 \u660e \u79c0 \u65f6\u95f4\uff1a2000\u5e749\u67089\u65e5 \u4e00\u3001\u6559\u6750\u5206\u6790\uff1a \u672c\u8282\u8bfe\u662f\u9ad8\u4e2d\u65b0\u6559\u6750\u300a\u6570\u5b66\u300b\u7b2c\u4e00\u518c\uff08\u4e0b\uff09\u00a74.8\u300a\u6b63\u5f26\u51fd\u6570\u3001\u4f59\u5f26\u51fd\u6570\u7684\u56fe\u8c61\u548c\u6027\u8d28\u300b \u7684\u7b2c\u4e00\u8282\uff0c\u662f\u5b66\u751f\u5728\u5df2\u638c\u63e1\u4e86\u4e00\u4e9b\u57fa\u672c\u51fd\u6570\u7684\u56fe\u8c61\u53ca\u5176\u753b\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u8fdb\u4e00\u6b65\u7814\u7a76\u4e09\u89d2\u51fd\u6570\u56fe\u8c61\u7684\u753b\u6cd5\uff0e\u4e3a\u4eca\u540e\u5b66\u4e60\u6b63\u5f26\u578b\u51fd\u6570 y\uff1dAsin (\u03c9x\uff0b\u03c6)\u7684\u56fe\u8c61\u53ca\u8fd0\u7528\u6570\u5f62\u7ed3\u5408\u601d\u60f3\u7814\u7a76\u6b63\u3001\u4f59\u5f26\u51fd\u6570\u7684\u6027\u8d28\u6253\u4e0b\u575a\u5b9e\u7684\u77e5\u8bc6\u57fa\u7840\uff0e\u56e0\u6b64\uff0c\u672c\u8282\u8bfe\u7684\u5185\u5bb9\u662f\u81f3\u5173\u91cd\u8981\u7684\uff0c\u5b83\u5bf9\u77e5\u8bc6\u7684\u638c\u63e1\u8d77\u5230\u4e86\u627f\u4e0a\u542f\u4e0b\u7684\u4f5c\u7528\uff0e\u4e8c\u3001\u5b66\u60c5\u5206\u6790\uff1a \u5728\u521d\u4e2d\u5b66\u751f\u5df2\u7ecf\u5b66\u4e60\u8fc7\u4e09\u6b65\u4f5c\u56fe\u6cd5\uff08\u5217\u8868\uff0c\u63cf\u70b9\u3001\u8fde\u7ebf\uff09??\u201c\u63cf\u70b9\u4f5c\u56fe\u201d\u6cd5\uff0c\u5bf9\u4e8e\u51fd\u6570y\uff1dsinx\uff0c\u5f53x\u53d6\u503c\u65f6\uff0cy\u7684\u503c\u5927\u90fd\u662f\u8fd1\u4f3c\u503c\uff0c\u52a0\u4e4b\u4f5c\u56fe\u4e0a\u7684\u8bef\u5dee\uff0c\u5f88\u96be\u8ba4\u8bc6\u65b0\u51fd\u6570y\uff1dsinx\u7684\u56fe\u8c61\u7684\u771f\u5b9e\u9762\u8c8c\u3002\u56e0\u4e3a\u5728\u524d\u9762\u5df2\u7ecf\u5b66\u4e60\u8fc7\u4e09\u89d2\u51fd\u6570\u7ebf\uff0c\u8fd9\u5c31\u4e3a\u7528\u51e0\u4f55\u6cd5\u4f5c\u56fe\u63d0\u4f9b\u4e86\u57fa\u7840\u3002\u52a8\u624b\u4f5c\u51fa\u51fd\u6570y\uff1dsinx\u548cy=cosx\u7684\u56fe\u8c61\uff0c\u5b66\u751f\u4e0d\u4f1a\u611f\u5230\u56f0\u96be\u3002\u4e09\u3001\u6559\u5b66\u76ee\u6807\uff1a\u4f9d\u636e\u6559\u5b66\u5927\u7eb2\u7684\u8981\u6c42\uff0c\u5236\u8ba2\u5982\u4e0b\u4e09\u7ef4\u6559\u5b66\u76ee\u6807\uff1a\u77e5\u8bc6\u76ee\u6807\u662f\uff1a1\uff0e\u7406\u89e3\u51e0\u4f55\u6cd5\u4f5c\u56fe\u539f\u7406\uff08\u96be\u70b9\uff09\uff1b 2\uff0e\u638c\u63e1\u4e94\u70b9\u6cd5\u4f5c\u56fe\uff08\u91cd\u70b9\uff09\uff1b 3\uff0e\u4e86\u89e3\u4e09\u89d2\u51fd\u6570\u56fe\u8c61\u7684\u53d8\u6362\u4f5c\u56fe\uff0e\u80fd\u529b\u76ee\u6807\u662f\uff1a\u901a\u8fc7\u8bc6\u8bb0\u6b63\u3001\u4f59\u5f26\u66f2\u7ebf\u7684\u5f62\u72b6\u7279\u5f81\uff0c\u57f9\u517b\u5b66\u751f\u5206\u6790\u95ee\u9898\u3001 \u89e3\u51b3\u95ee\u9898\u7684\u80fd\u529b\uff1b\u5f3a\u5316\u5b66\u751f\uff02\u6570\u5f62\u7ed3\u5408\uff02\u7684\u6570\u5b66\u601d\u60f3\uff0e\u53d1\u5c55\u76ee\u6807\u662f\uff1a\u6559\u7ed9\u5b66\u751f\u7075\u6d3b\u7684\u601d\u7ef4\u65b9\u6cd5\uff0c\u57f9\u517b\u5b66\u751f\u7684\u5b66\u4e60\u5174\u8da3\u548c\u52c7\u4e8e \u63a2\u7d22\u3001\u52c7\u4e8e\u521b\u65b0\u7684\u7cbe\u795e\uff0c\u63d0\u9ad8\u7efc\u5408\u7d20\u8d28\uff0e\u56db\u3001\u8bbe\u8ba1\u7406\u5ff5\uff1a 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\u4e09\u89d2\u51fd\u6570\u56fe\u50cf\u4e0e\u6027\u8d28
\u4e00 \u3001\u77e5\u8bc6\u70b9\u5f52\u7eb3 \uff08\u4e09\u89d2\u51fd\u6570\u7684\u56fe\u50cf\u4e0e\u6027\u8d28\uff09
1\u4e09\u89d2\u51fd\u6570\u7684\u56fe\u50cf
\uff08\u7565\uff09
2.\u4e09\u89d2\u51fd\u6570\u7684\u6027\u8d28
\uff081\uff09\u4e09\u89d2\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u3001\u503c\u57df\u3001\u6700\u503c\u7b49
\u51fd\u6570 \u5b9a\u4e49\u57df \u503c\u57df \u5468\u671f
Y=sin x R \uff3b-1,1\uff3d 2\u03a0
Y=cos x R \uff3b-1,1\uff3d 2\u03a0
Y=tan x \uff5bx/x\u2260kx+\u03a0/2,kz\uff5d R \u03a0
\u5b9a\u4e49\u57df\uff1a\u5728\u6570\u5b66\u4e2d\u53ef\u4ee5\u88ab\u770b\u4f5c\u4e3a\u51fd\u6570\u7684\u6240\u6709\u8f93\u5165\u503c\u7684\u96c6\u5408\u3002
\u51fd\u6570\u5b9a\u4e49\u57df\u7684\u4e09\u7c7b\u6c42\u6cd5
\u3000\u3000\u4e00\u3001\u7ed9\u51fa\u51fd\u6570\u89e3\u6790\u5f0f\u6c42\u5176\u5b9a\u4e49\u57df\uff0c\u4e00\u822c\u662f\u5148\u5217\u51fa\u9650\u5236\u6761\u4ef6\u7684\u4e0d\u7b49\u5f0f\uff08\u7ec4\uff09\uff0c\u518d\u8fdb\u884c\u6c42\u89e3\u3002
\u3000\u3000\u4e8c. \u7ed9\u51fa\u51fd\u6570\u7684\u5b9a\u4e49\u57df\uff0c\u6c42\u51fd\u6570\u7684\u5b9a\u4e49\u57df\uff0c\u5176\u89e3\u6cd5\u6b65\u9aa4\u662f\uff1a\u82e5\u5df2\u77e5\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u4e3a\uff0c\u5219\u5176\u590d\u5408\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u5e94\u7531\u4e0d\u7b49\u5f0f\u89e3\u5f97\u3002
\u3000\u3000\u4e09. \u7ed9\u51fa\u7684\u5b9a\u4e49\u57df\uff0c\u6c42\u7684\u5b9a\u4e49\u57df\uff0c\u5176\u89e3\u6cd5\u6b65\u9aa4\u662f\uff1a\u82e5\u5df2\u77e5\u7684\u5b9a\u4e49\u57df\u4e3a\uff0c\u5219\u7684\u5b9a\u4e49\u57df\u662f\u5728\u65f6\u7684\u53d6\u503c\u8303\u56f4\u3002
\u3000\u503c\u57df\uff1a\u51fd\u6570\u4e2d\uff0c\u5e94\u53d8\u91cf\u7684\u53d6\u503c\u8303\u56f4\u53eb\u505a\u8fd9\u4e2a\u51fd\u6570\u7684\u503c\u57df,\u5728\u6570\u5b66\u4e2d\u662f\u51fd\u6570\u5728\u5b9a\u4e49\u57df\u4e2d\u5e94\u53d8\u91cf\u6240\u6709\u503c\u7684\u96c6\u5408
[\u7f16\u8f91\u672c\u6bb5]\u5e38\u7528\u7684\u6c42\u503c\u57df\u7684\u65b9\u6cd5
\u3000\u3000\uff081\uff09\u5316\u5f52\u6cd5\uff1b\uff082\uff09\u56fe\u8c61\u6cd5\uff08\u6570\u5f62\u7ed3\u5408\uff09\uff0c
\u3000\u3000\uff083\uff09\u51fd\u6570\u5355\u8c03\u6027\u6cd5\uff0c
\u3000\u3000\uff084\uff09\u914d\u65b9\u6cd5\uff0c\uff085\uff09\u6362\u5143\u6cd5\uff0c\uff086\uff09\u53cd\u51fd\u6570\u6cd5\uff08\u9006\u6c42\u6cd5\uff09\uff0c\uff087\uff09\u5224\u522b\u5f0f\u6cd5\uff0c\uff088\uff09\u590d\u5408\u51fd\u6570\u6cd5\uff0c\uff089\uff09\u4e09\u89d2\u4ee3\u6362\u6cd5\uff0c\uff0810\uff09\u57fa\u672c\u4e0d\u7b49\u5f0f\u6cd5\u7b49
\u5468\u671f\uff1a\u51fd\u6570f\uff08x\uff09\u7684\u6700\u5c0f\u6b63\u5468\u671fT\u5fc5\u987b\u6ee1\u8db3\u4e00\u4e0b\u4e24\u4e2a\u6761\u4ef6\uff1a\uff081\uff09\u5f53x\u53bb\u5b9a\u4e49\u57df\u5185\u7684\u6bcf\u4e00\u4e2a\u503c\u65f6\uff0c\u90fd\u6709f\uff08x+T\uff09=f\uff08x\uff09\u3002\uff082\uff09T\u65f6\u4e0d\u4e3a\u96f6\u7684\u6700\u5c0f\u6b63\u6570\uff0c\u4e00\u822c\u5730\uff0c\u82e5T\u4e3af\uff08x\uff09\u7684\u5468\u671f\uff0c\u5219nT\u4e5f\u4e3af\uff08x\uff09\u6240\u8c13\u5468\u671f\uff0c\u5373f\uff08x\uff09=f\uff08x+nT\uff09\u3002
\uff082\uff09\u4e09\u89d2\u51fd\u6570\u7684\u5947\u5076\u6027\u4e0e\u5355\u8c03\u6027
\u51fd\u6570 Y=sin x Y=cos x Y=tan x
\u5947\u5076\u6027 \u5947 \u5076 \u5947
\uff081\uff09\u5947\u5076\u6027
1\uff09\u4e3a\u5947\u51fd\u6570\uff082\uff09\u4e3a\u5076\u51fd\u6570
\u3000\u3000\u5bf9\u4e8e\u51fd\u6570f(x)
\u3000\u3000\uff081\uff09\u5982\u679c\u5bf9\u4e8e\u51fd\u6570\u5b9a\u4e49\u57df\u5185\u7684\u4efb\u610f\u4e00\u4e2ax\uff0c\u90fd\u6709f(-x)=\uff0df(x)\uff0c\u90a3\u4e48\u51fd\u6570f(x)\u5c31\u53eb\u505a\u5947\u51fd\u6570\u3002
\u3000\u3000\uff082\uff09\u5982\u679c\u5bf9\u4e8e\u51fd\u6570\u5b9a\u4e49\u57df\u5185\u7684\u4efb\u610f\u4e00\u4e2ax\uff0c\u90fd\u6709f(-x)=f(x)\uff0c\u90a3\u4e48\u51fd\u6570f(x)\u5c31\u53eb\u505a\u5076\u51fd\u6570\u3002
\u3000\u3000\uff083\uff09\u5982\u679c\u5bf9\u4e8e\u51fd\u6570\u5b9a\u4e49\u57df\u5185\u7684\u4efb\u610f\u4e00\u4e2ax\uff0cf(-x)=-f(x)\u4e0ef(-x)=f(x)\u540c\u65f6\u6210\u7acb\uff0c\u90a3\u4e48\u51fd\u6570f(x)\u65e2\u662f\u5947\u51fd\u6570\u53c8\u662f\u5076\u51fd\u6570\uff0c\u79f0\u4e3a\u65e2\u5947\u53c8\u5076\u51fd\u6570\u3002
\u3000\u3000\uff084\uff09\u5982\u679c\u5bf9\u4e8e\u51fd\u6570\u5b9a\u4e49\u57df\u5185\u7684\u4efb\u610f\u4e00\u4e2ax\uff0cf(-x)=-f(x)\u6216f(-x)=f(x)\u90fd\u4e0d\u80fd\u6210\u7acb\uff0c\u90a3\u4e48\u51fd\u6570f(x)\u65e2\u4e0d\u662f\u5947\u51fd\u6570\u53c8\u4e0d\u662f\u5076\u51fd\u6570\uff0c\u79f0\u4e3a\u975e\u5947\u975e\u5076\u51fd\u6570\u3002
\u3000\u3000\u8bf4\u660e\uff1a\u2460\u5947\u3001\u5076\u6027\u662f\u51fd\u6570\u7684\u6574\u4f53\u6027\u8d28\uff0c\u5bf9\u6574\u4e2a\u5b9a\u4e49\u57df\u800c\u8a00
\u3000\u3000\u2461\u5947\u3001\u5076\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u4e00\u5b9a\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\uff0c\u5982\u679c\u4e00\u4e2a\u51fd\u6570\u7684\u5b9a\u4e49\u57df\u4e0d\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\uff0c\u5219\u8fd9\u4e2a\u51fd\u6570\u4e00\u5b9a\u4e0d\u662f\u5947\uff08\u6216\u5076\uff09\u51fd\u6570\u3002
\u3000\u3000\u2462\u5224\u65ad\u6216\u8bc1\u660e\u51fd\u6570\u662f\u5426\u5177\u6709\u5947\u5076\u6027\u7684\u6839\u636e\u662f\u5b9a\u4e49
\uff082\uff09\u5355\u8c03\u6027
\u82e5\u51fd\u6570y=f(x)\u5728\u67d0\u4e2a\u533a\u95f4\u662f\u589e\u51fd\u6570\u6216\u51cf\u51fd\u6570,\u5219\u5c31\u8bf4\u51fd\u6570\u5728\u8fd9\u4e00\u533a\u95f4\u5177\u6709(\u4e25\u683c\u7684)\u5355\u8c03\u6027,\u8fd9\u4e00\u533a\u95f4\u53eb\u505a\u51fd\u6570\u7684\u5355\u8c03\u533a\u95f4.\u6b64\u65f6\u4e5f\u8bf4\u51fd\u6570\u662f\u8fd9\u4e00\u533a\u95f4\u4e0a\u7684\u5355\u8c03\u51fd\u6570\u3002
\u3000\u3000\u5728\u5355\u8c03\u533a\u95f4\u4e0a,\u589e\u51fd\u6570\u7684\u56fe\u50cf\u662f\u4e0a\u5347\u7684,\u51cf\u51fd\u6570\u7684\u56fe\u50cf\u662f\u4e0b\u964d\u7684\u3002
\u7ec3\u4e60
1.\u82e5\u51fd\u6570y=sin(2x+\u03c6)\u4e3a\u5076\u51fd\u6570\uff0c\u5219\u03c6\u7684\u4e00\u4e2a\u503c\u662f\uff08\uff09A.\u03c6=-\u03c0 B.\u03c6=-\u03c0/2 C.\u03c6=2\u03c0 D.\u03c6=\u03c0/42.\u51fd\u6570f(x)=sin(\u03c9x+\u03c6)cos(\u03c9x+\u03c6),(\u03c9>0)\u4ee52\u4e3a\u6700\u5c0f\u6b63\u5468\u671f\uff0c\u4e14\u5728x=2\u65f6\u53d6\u6700\u5927\u503c\uff0c\u5219\u03c6\u7684\u4e00\u4e2a\u503c\u662f\uff08\uff09A.7/4\u03c0 B.-5/4\u03c0 C.-3/4\u03c0 D.\u03c0/2
3 \u8bbe\u4e8c\u6b21\u51fd\u6570f(x)=x2+bx+c(b,c\u2208R),\u5df2\u77e5\u4e0d\u8bba\u03b1\u3001\u03b2\u4e3a\u4f55\u5b9e\u6570\u6052\u6709f(sin\u03b1)\u22650\u548cf(2+cos\u03b2)\u22640
(1)\u6c42\u8bc1 b+c=\uff0d1\uff1b
(2)\u6c42\u8bc1c\u22653\uff1b
(3)\u82e5\u51fd\u6570f(sin\u03b1)\u7684\u6700\u5927\u503c\u4e3a8\uff0c\u6c42b\uff0cc\u7684\u503c
\u53c2\u8003\u7b54\u6848
1B
2.C
3 \u89e3 (1)\u2235\uff0d1\u2264sin\u03b1\u22641\u4e14f(sin\u03b1)\u22650\u6052\u6210\u7acb\uff0c\u2234f(1)\u22650
\u22351\u22642+cos\u03b2\u22643,\u4e14f(2+cos\u03b2)\u22640\u6052\u6210\u7acb \u2234f(1)\u22640
\u4ece\u800c\u77e5f(1)=0\u2234b+c+1=0
(2)\u7531f(2+cos\u03b2)\u22640,\u77e5f(3)\u22640,\u22349+3b+c\u22640 \u53c8\u56e0\u4e3ab+c=\uff0d1,\u2234c\u22653
(3) b=\uff0d4,c=3
\u9ad8\u4e2d\u5b66\u4e60\u7f51 www.90house.cn \u539f\u6587\u94fe\u63a5\uff1ahttp://www.90house.cn/gaozhongshuxuezt/1546.html
\u4e09\u89d2\u51fd\u6570\u516c\u5f0f
\u4e24\u89d2\u548c\u516c\u5f0f
sin(A+B) = sinAcosB+cosAsinB
sin(A-B) = sinAcosB-cosAsinB
cos(A+B) = cosAcosB-sinAsinB
cos(A-B) = cosAcosB+sinAsinB
tan(A+B) = (tanA+tanB)/(1-tanAtanB)
tan(A-B) = (tanA-tanB)/(1+tanAtanB)
cot(A+B) = (cotAcotB-1)/(cotB+cotA)
cot(A-B) = (cotAcotB+1)/(cotB-cotA)
\u500d\u89d2\u516c\u5f0f
tan2A = 2tanA/(1-tan^2 A)
Sin2A=2SinA•CosA
Cos2A = Cos^2 A--Sin^2 A
=2Cos^2 A\u20141
=1\u20142sin^2 A
\u4e09\u500d\u89d2\u516c\u5f0f
sin3A = 3sinA-4(sinA)^3;
cos3A = 4(cosA)^3 -3cosA
tan3a = tan a • tan(\u03c0/3+a)• tan(\u03c0/3-a)
\u534a\u89d2\u516c\u5f0f
sin(A/2) = \u221a{(1--cosA)/2}
cos(A/2) = \u221a{(1+cosA)/2}
tan(A/2) = \u221a{(1--cosA)/(1+cosA)}
cot(A/2) = \u221a{(1+cosA)/(1-cosA)}
tan(A/2) = (1--cosA)/sinA=sinA/(1+cosA)
\u548c\u5dee\u5316\u79ef
sin(a)+sin(b) = 2sin[(a+b)/2]cos[(a-b)/2]
sin(a)-sin(b) = 2cos[(a+b)/2]sin[(a-b)/2]
cos(a)+cos(b) = 2cos[(a+b)/2]cos[(a-b)/2]
cos(a)-cos(b) = -2sin[(a+b)/2]sin[(a-b)/2]
tanA+tanB=sin(A+B)/cosAcosB
\u79ef\u5316\u548c\u5dee
sin(a)sin(b) = -1/2*[cos(a+b)-cos(a-b)]
cos(a)cos(b) = 1/2*[cos(a+b)+cos(a-b)]
sin(a)cos(b) = 1/2*[sin(a+b)+sin(a-b)]
cos(a)sin(b) = 1/2*[sin(a+b)-sin(a-b)]
\u8bf1\u5bfc\u516c\u5f0f
sin(-a) = -sin(a)
cos(-a) = cos(a)
sin(\u03c0/2-a) = cos(a)
cos(\u03c0/2-a) = sin(a)
sin(\u03c0/2+a) = cos(a)
cos(\u03c0/2+a) = -sin(a)
sin(\u03c0-a) = sin(a)
cos(\u03c0-a) = -cos(a)
sin(\u03c0+a) = -sin(a)
cos(\u03c0+a) = -cos(a)
tgA=tanA = sinA/cosA
\u4e07\u80fd\u516c\u5f0f
sin(a) = [2tan(a/2)] / {1+[tan(a/2)]^2}
cos(a) = {1-[tan(a/2)]^2} / {1+[tan(a/2)]^2}
tan(a) = [2tan(a/2)]/{1-[tan(a/2)]^2}
\u5176\u5b83\u516c\u5f0f
a•sin(a)+b•cos(a) = [\u221a(a^2+b^2)]*sin(a+c) [\u5176\u4e2d\uff0ctan(c)=b/a]
a•sin(a)-b•cos(a) = [\u221a(a^2+b^2)]*cos(a-c) [\u5176\u4e2d\uff0ctan(c)=a/b]
1+sin(a) = [sin(a/2)+cos(a/2)]^2;
1-sin(a) = [sin(a/2)-cos(a/2)]^2;;
\u5176\u4ed6\u975e\u91cd\u70b9\u4e09\u89d2\u51fd\u6570
csc(a) = 1/sin(a)
sec(a) = 1/cos(a)
\u53cc\u66f2\u51fd\u6570
sinh(a) = [e^a-e^(-a)]/2
cosh(a) = [e^a+e^(-a)]/2
tg h(a) = sin h(a)/cos h(a)
\u516c\u5f0f\u4e00\uff1a
\u8bbe\u03b1\u4e3a\u4efb\u610f\u89d2\uff0c\u7ec8\u8fb9\u76f8\u540c\u7684\u89d2\u7684\u540c\u4e00\u4e09\u89d2\u51fd\u6570\u7684\u503c\u76f8\u7b49\uff1a
sin\uff082k\u03c0\uff0b\u03b1\uff09= sin\u03b1
cos\uff082k\u03c0\uff0b\u03b1\uff09= cos\u03b1
tan\uff082k\u03c0\uff0b\u03b1\uff09= tan\u03b1
cot\uff082k\u03c0\uff0b\u03b1\uff09= cot\u03b1
\u516c\u5f0f\u4e8c\uff1a
\u8bbe\u03b1\u4e3a\u4efb\u610f\u89d2\uff0c\u03c0+\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e0e\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
sin\uff08\u03c0\uff0b\u03b1\uff09= -sin\u03b1
cos\uff08\u03c0\uff0b\u03b1\uff09= -cos\u03b1
tan\uff08\u03c0\uff0b\u03b1\uff09= tan\u03b1
cot\uff08\u03c0\uff0b\u03b1\uff09= cot\u03b1
\u516c\u5f0f\u4e09\uff1a
\u4efb\u610f\u89d2\u03b1\u4e0e -\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
sin\uff08-\u03b1\uff09= -sin\u03b1
cos\uff08-\u03b1\uff09= cos\u03b1
tan\uff08-\u03b1\uff09= -tan\u03b1
cot\uff08-\u03b1\uff09= -cot\u03b1
\u516c\u5f0f\u56db\uff1a
\u5229\u7528\u516c\u5f0f\u4e8c\u548c\u516c\u5f0f\u4e09\u53ef\u4ee5\u5f97\u5230\u03c0-\u03b1\u4e0e\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
sin\uff08\u03c0-\u03b1\uff09= sin\u03b1
cos\uff08\u03c0-\u03b1\uff09= -cos\u03b1
tan\uff08\u03c0-\u03b1\uff09= -tan\u03b1
cot\uff08\u03c0-\u03b1\uff09= -cot\u03b1
\u516c\u5f0f\u4e94\uff1a
\u5229\u7528\u516c\u5f0f-\u548c\u516c\u5f0f\u4e09\u53ef\u4ee5\u5f97\u52302\u03c0-\u03b1\u4e0e\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
sin\uff082\u03c0-\u03b1\uff09= -sin\u03b1
cos\uff082\u03c0-\u03b1\uff09= cos\u03b1
tan\uff082\u03c0-\u03b1\uff09= -tan\u03b1
cot\uff082\u03c0-\u03b1\uff09= -cot\u03b1
\u516c\u5f0f\u516d\uff1a
\u03c0/2\u00b1\u03b1\u53ca3\u03c0/2\u00b1\u03b1\u4e0e\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
sin\uff08\u03c0/2+\u03b1\uff09= cos\u03b1
cos\uff08\u03c0/2+\u03b1\uff09= -sin\u03b1
tan\uff08\u03c0/2+\u03b1\uff09= -cot\u03b1
cot\uff08\u03c0/2+\u03b1\uff09= -tan\u03b1
sin\uff08\u03c0/2-\u03b1\uff09= cos\u03b1
cos\uff08\u03c0/2-\u03b1\uff09= sin\u03b1
tan\uff08\u03c0/2-\u03b1\uff09= cot\u03b1
cot\uff08\u03c0/2-\u03b1\uff09= tan\u03b1
sin\uff083\u03c0/2+\u03b1\uff09= -cos\u03b1
cos\uff083\u03c0/2+\u03b1\uff09= sin\u03b1
tan\uff083\u03c0/2+\u03b1\uff09= -cot\u03b1
cot\uff083\u03c0/2+\u03b1\uff09= -tan\u03b1
sin\uff083\u03c0/2-\u03b1\uff09= -cos\u03b1
cos\uff083\u03c0/2-\u03b1\uff09= -sin\u03b1
tan\uff083\u03c0/2-\u03b1\uff09= cot\u03b1
cot\uff083\u03c0/2-\u03b1\uff09= tan\u03b1
(\u4ee5\u4e0ak\u2208Z)
\u4e09\u89d2\u51fd\u6570\u516c\u5f0f\u5927\u5168
\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u516c\u5f0f
sin \u03b1=\u2220\u03b1\u7684\u5bf9\u8fb9 / \u659c\u8fb9
cos \u03b1=\u2220\u03b1\u7684\u90bb\u8fb9 / \u659c\u8fb9
tan \u03b1=\u2220\u03b1\u7684\u5bf9\u8fb9 / \u2220\u03b1\u7684\u90bb\u8fb9
cot \u03b1=\u2220\u03b1\u7684\u90bb\u8fb9 / \u2220\u03b1\u7684\u5bf9\u8fb9
\u500d\u89d2\u516c\u5f0f
Sin2A=2SinA?CosA
Cos2A=CosA^2-SinA^2=1-2SinA^2=2CosA^2-1
tan2A=\uff082tanA\uff09/\uff081-tanA^2\uff09
\uff08\u6ce8\uff1aSinA^2 \u662fsinA\u7684\u5e73\u65b9 sin2\uff08A\uff09 \uff09
\u4e09\u500d\u89d2\u516c\u5f0f
sin3\u03b1=4sin\u03b1•sin(\u03c0/3+\u03b1)sin(\u03c0/3-\u03b1)
cos3\u03b1=4cos\u03b1•cos(\u03c0/3+\u03b1)cos(\u03c0/3-\u03b1)
tan3a = tan a • tan(\u03c0/3+a)• tan(\u03c0/3-a)
\u4e09\u500d\u89d2\u516c\u5f0f\u63a8\u5bfc
sin3a
=sin(2a+a)
=sin2acosa+cos2asina
\u8f85\u52a9\u89d2\u516c\u5f0f
Asin\u03b1+Bcos\u03b1=(A^2+B^2)^(1/2)sin(\u03b1+t)\uff0c\u5176\u4e2d
sint=B/(A^2+B^2)^(1/2)
cost=A/(A^2+B^2)^(1/2)
tant=B/A
Asin\u03b1+Bcos\u03b1=(A^2+B^2)^(1/2)cos(\u03b1-t)\uff0ctant=A/B
\u964d\u5e42\u516c\u5f0f
sin^2(\u03b1)=(1-cos(2\u03b1))/2=versin(2\u03b1)/2
cos^2(\u03b1)=(1+cos(2\u03b1))/2=covers(2\u03b1)/2
tan^2(\u03b1)=(1-cos(2\u03b1))/(1+cos(2\u03b1))
\u63a8\u5bfc\u516c\u5f0f
tan\u03b1+cot\u03b1=2/sin2\u03b1
tan\u03b1-cot\u03b1=-2cot2\u03b1
1+cos2\u03b1=2cos^2\u03b1
1-cos2\u03b1=2sin^2\u03b1
1+sin\u03b1=(sin\u03b1/2+cos\u03b1/2)^2
=2sina(1-sin²a)+(1-2sin²a)sina
=3sina-4sin³a
cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos²a-1)cosa-2(1-sin²a)cosa
=4cos³a-3cosa
sin3a=3sina-4sin³a
=4sina(3/4-sin²a)
=4sina[(\u221a3/2)²-sin²a]
=4sina(sin²60\u00b0-sin²a)
=4sina(sin60\u00b0+sina)(sin60\u00b0-sina)
=4sina*2sin[(60+a)/2]cos[(60\u00b0-a)/2]*2sin[(60\u00b0-a)/2]cos[(60\u00b0-a)/2]
=4sinasin(60\u00b0+a)sin(60\u00b0-a)
cos3a=4cos³a-3cosa
=4cosa(cos²a-3/4)
=4cosa[cos²a-(\u221a3/2)²]
=4cosa(cos²a-cos²30\u00b0)
=4cosa(cosa+cos30\u00b0)(cosa-cos30\u00b0)
=4cosa*2cos[(a+30\u00b0)/2]cos[(a-30\u00b0)/2]*{-2sin[(a+30\u00b0)/2]sin[(a-30\u00b0)/2]}
=-4cosasin(a+30\u00b0)sin(a-30\u00b0)
=-4cosasin[90\u00b0-(60\u00b0-a)]sin[-90\u00b0+(60\u00b0+a)]
=-4cosacos(60\u00b0-a)[-cos(60\u00b0+a)]
=4cosacos(60\u00b0-a)cos(60\u00b0+a)
\u4e0a\u8ff0\u4e24\u5f0f\u76f8\u6bd4\u53ef\u5f97
tan3a=tanatan(60\u00b0-a)tan(60\u00b0+a)
\u534a\u89d2\u516c\u5f0f
tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA);
cot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA.
sin^2(a/2)=(1-cos(a))/2
cos^2(a/2)=(1+cos(a))/2
tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))
\u4e09\u89d2\u548c
sin(\u03b1+\u03b2+\u03b3)=sin\u03b1•cos\u03b2•cos\u03b3+cos\u03b1•sin\u03b2•cos\u03b3+cos\u03b1•cos\u03b2•sin\u03b3-sin\u03b1•sin\u03b2•sin\u03b3
cos(\u03b1+\u03b2+\u03b3)=cos\u03b1•cos\u03b2•cos\u03b3-cos\u03b1•sin\u03b2•sin\u03b3-sin\u03b1•cos\u03b2•sin\u03b3-sin\u03b1•sin\u03b2•cos\u03b3
tan(\u03b1+\u03b2+\u03b3)=(tan\u03b1+tan\u03b2+tan\u03b3-tan\u03b1•tan\u03b2•tan\u03b3)/(1-tan\u03b1•tan\u03b2-tan\u03b2•tan\u03b3-tan\u03b3•tan\u03b1)
\u4e24\u89d2\u548c\u5dee
cos(\u03b1+\u03b2)=cos\u03b1•cos\u03b2-sin\u03b1•sin\u03b2
cos(\u03b1-\u03b2)=cos\u03b1•cos\u03b2+sin\u03b1•sin\u03b2
sin(\u03b1\u00b1\u03b2)=sin\u03b1•cos\u03b2\u00b1cos\u03b1•sin\u03b2
tan(\u03b1+\u03b2)=(tan\u03b1+tan\u03b2)/(1-tan\u03b1•tan\u03b2)
tan(\u03b1-\u03b2)=(tan\u03b1-tan\u03b2)/(1+tan\u03b1•tan\u03b2)
\u548c\u5dee\u5316\u79ef
sin\u03b8+sin\u03c6 = 2 sin[(\u03b8+\u03c6)/2] cos[(\u03b8-\u03c6)/2]
sin\u03b8-sin\u03c6 = 2 cos[(\u03b8+\u03c6)/2] sin[(\u03b8-\u03c6)/2]
cos\u03b8+cos\u03c6 = 2 cos[(\u03b8+\u03c6)/2] cos[(\u03b8-\u03c6)/2]
cos\u03b8-cos\u03c6 = -2 sin[(\u03b8+\u03c6)/2] sin[(\u03b8-\u03c6)/2]
tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB)
tanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)
\u79ef\u5316\u548c\u5dee
sin\u03b1sin\u03b2 = [cos(\u03b1-\u03b2)-cos(\u03b1+\u03b2)] /2
cos\u03b1cos\u03b2 = [cos(\u03b1+\u03b2)+cos(\u03b1-\u03b2)]/2
sin\u03b1cos\u03b2 = [sin(\u03b1+\u03b2)+sin(\u03b1-\u03b2)]/2
cos\u03b1sin\u03b2 = [sin(\u03b1+\u03b2)-sin(\u03b1-\u03b2)]/2
\u8bf1\u5bfc\u516c\u5f0f
sin(-\u03b1) = -sin\u03b1
cos(-\u03b1) = cos\u03b1
tan (\u2014a)=-tan\u03b1
sin(\u03c0/2-\u03b1) = cos\u03b1
cos(\u03c0/2-\u03b1) = sin\u03b1
sin(\u03c0/2+\u03b1) = cos\u03b1
cos(\u03c0/2+\u03b1) = -sin\u03b1
sin(\u03c0-\u03b1) = sin\u03b1
cos(\u03c0-\u03b1) = -cos\u03b1
sin(\u03c0+\u03b1) = -sin\u03b1
cos(\u03c0+\u03b1) = -cos\u03b1
tanA= sinA/cosA
tan\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
tan\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
tan\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
tan\uff08\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
\u8bf1\u5bfc\u516c\u5f0f\u8bb0\u80cc\u8bc0\u7a8d\uff1a\u5947\u53d8\u5076\u4e0d\u53d8\uff0c\u7b26\u53f7\u770b\u8c61\u9650
\u4e07\u80fd\u516c\u5f0f
sin\u03b1=2tan(\u03b1/2\uff09/\u30141+tan\uff3e(\u03b1/2)\u3015
cos\u03b1=\u30141-tan\uff3e(\u03b1/2)\u3015/1+tan\uff3e(\u03b1/2)\u3015
tan\u03b1=2tan(\u03b1/2)/\u30141-tan\uff3e(\u03b1/2)\u3015
\u5176\u5b83\u516c\u5f0f
(1)(sin\u03b1)^2+(cos\u03b1)^2=1
(2)1+(tan\u03b1)^2=(sec\u03b1)^2
(3)1+(cot\u03b1)^2=(csc\u03b1)^2
\u8bc1\u660e\u4e0b\u9762\u4e24\u5f0f,\u53ea\u9700\u5c06\u4e00\u5f0f,\u5de6\u53f3\u540c\u9664(sin\u03b1)^2,\u7b2c\u4e8c\u4e2a\u9664(cos\u03b1)^2\u5373\u53ef
(4)\u5bf9\u4e8e\u4efb\u610f\u975e\u76f4\u89d2\u4e09\u89d2\u5f62,\u603b\u6709
tanA+tanB+tanC=tanAtanBtanC
\u8bc1:
A+B=\u03c0-C
tan(A+B)=tan(\u03c0-C)
(tanA+tanB)/(1-tanAtanB)=(tan\u03c0-tanC)/(1+tan\u03c0tanC)
\u6574\u7406\u53ef\u5f97
tanA+tanB+tanC=tanAtanBtanC
\u5f97\u8bc1
\u540c\u6837\u53ef\u4ee5\u5f97\u8bc1,\u5f53x+y+z=n\u03c0(n\u2208Z)\u65f6,\u8be5\u5173\u7cfb\u5f0f\u4e5f\u6210\u7acb
\u7531tanA+tanB+tanC=tanAtanBtanC\u53ef\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba
(5)cotAcotB+cotAcotC+cotBcotC=1
(6)cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2)
(7)(cosA\uff09^2+(cosB\uff09^2+(cosC\uff09^2=1-2cosAcosBcosC
(8)\uff08sinA\uff09^2+\uff08sinB\uff09^2+\uff08sinC\uff09^2=2+2cosAcosBcosC
(9)sin\u03b1+sin(\u03b1+2\u03c0/n)+sin(\u03b1+2\u03c0*2/n)+sin(\u03b1+2\u03c0*3/n)+\u2026\u2026+sin[\u03b1+2\u03c0*(n-1)/n]=0
cos\u03b1+cos(\u03b1+2\u03c0/n)+cos(\u03b1+2\u03c0*2/n)+cos(\u03b1+2\u03c0*3/n)+\u2026\u2026+cos[\u03b1+2\u03c0*(n-1)/n]=0 \u4ee5\u53ca
sin^2(\u03b1)+sin^2(\u03b1-2\u03c0/3)+sin^2(\u03b1+2\u03c0/3)=3/2
tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0
\u4e00,\u8bf1\u5bfc\u516c\u5f0f
\u53e3\u8bc0:(\u5206\u5b50)\u5947\u53d8\u5076\u4e0d\u53d8,\u7b26\u53f7\u770b\u8c61\u9650.
1. sin (\u03b1+k•360)=sin \u03b1
cos (\u03b1+k•360)=cos a
tan (\u03b1+k•360)=tan \u03b1
2. sin(180\u00b0+\u03b2)=-sin\u03b1
cos(180\u00b0+\u03b2)=-cosa
3. sin(-\u03b1)=-sina
cos(-a)=cos\u03b1
4*. tan(180\u00b0+\u03b1)=tan\u03b1
tan(-\u03b1)=tan\u03b1
5. sin(180\u00b0-\u03b1)=sin\u03b1
cos(180\u00b0-\u03b1)=-cos\u03b1
6. sin(360\u00b0-\u03b1)=-sin\u03b1
cos(360\u00b0-\u03b1)=cos\u03b1
7. sin(\u03c0/2-\u03b1)=cos\u03b1
cos(\u03c0/2-\u03b1)=sin\u03b1
8*. Sin(3\u03c0/2-\u03b1)=-cos\u03b1
cos(3\u03c0/2-\u03b1)=-sin\u03b1
9*. Sin(\u03c0/2+\u03b1)=cos\u03b1
cos(\u03c0/2+a)=-sin\u03b1
10*.sin(3\u03c0/2+\u03b1)=-cos\u03b1
cos(3\u03c0/2+\u03b1)=sin\u03b1
\u4e8c,\u4e24\u89d2\u548c\u4e0e\u5dee\u7684\u4e09\u89d2\u51fd\u6570
1. \u4e24\u70b9\u8ddd\u79bb\u516c\u5f0f
2. S(\u03b1+\u03b2): sin(\u03b1+\u03b2)=sin\u03b1cos\u03b2+cos\u03b1sin\u03b2
C(\u03b1+\u03b2): cos(\u03b1+\u03b2)=cos\u03b1cos\u03b2-sin\u03b1sin\u03b2
3. S(\u03b1-\u03b2): sin(\u03b1-\u03b2)=sin\u03b1cos\u03b2-cos\u03b1sin\u03b2
C(\u03b1-\u03b2): cos(\u03b1-\u03b2)=cos\u03b1cos\u03b2+sin\u03b1sin\u03b2
4. T(\u03b1+\u03b2):
T(\u03b1-\u03b2):
5*.
\u4e09,\u4e8c\u500d\u89d2\u516c\u5f0f
1. S2\u03b1: sin2\u03b1=2sin\u03b1cos\u03b1
2. C2a: cos2\u03b1=cos2\u03b1-sin2a
3. T2\u03b1: tan2\u03b1=(2tan\u03b1)/(1-tan2\u03b1)
4. C2a': cos2\u03b1=1-2sin2\u03b1
cos2\u03b1=2cos2\u03b1-1
\u56db*,\u5176\u5b83\u6742\u9879(\u5168\u90e8\u4e0d\u53ef\u76f4\u63a5\u7528)
1.\u8f85\u52a9\u89d2\u516c\u5f0f
asin\u03b1+bcos\u03b1=sin(a+\u03c6),\u5176\u4e2dtan\u03c6=b/a,\u5176\u7ec8\u8fb9\u8fc7\u70b9(a, b)
asin\u03b1+bcos\u03b1=cos(a-\u03c6),\u5176\u4e2dtan\u03c6=a/b,\u5176\u7ec8\u8fb9\u8fc7\u70b9(b,a)
2.\u964d\u6b21,\u914d\u65b9\u516c\u5f0f
\u964d\u6b21:
sin2\u03b8=(1-cos2\u03b8)/2
cos2\u03b8=(1+cos2\u03b8)/2
\u914d\u65b9
1\u00b1sin\u03b8=[sin(\u03b8/2)\u00b1cos(\u03b8/2)]2
1+cos\u03b8=2cos2(\u03b8/2)
1-cos\u03b8=2sin2(\u03b8/2)
3. \u4e09\u500d\u89d2\u516c\u5f0f
sin3\u03b8=3sin\u03b8-4sin3\u03b8
cos3\u03b8=4cos3-3cos\u03b8
4. \u4e07\u80fd\u516c\u5f0f
5. \u548c\u5dee\u5316\u79ef\u516c\u5f0f
sin\u03b1+sin\u03b2=
sin\u03b1-sin\u03b2=
cos\u03b1+cos\u03b2=
cos\u03b1-cos\u03b2=
6. \u79ef\u5316\u548c\u5dee\u516c\u5f0f
sin\u03b1sin\u03b2=1/2[sin(\u03b1+\u03b2)+sin(\u03b1-\u03b2)]
cos\u03b1sin\u03b2=1/2[sin(\u03b1+\u03b2)-sin(\u03b1-\u03b2)]
sin\u03b1sin\u03b2-1/2[cos(\u03b1+\u03b2)-cos(\u03b1-\u03b2)]
cos\u03b1cos\u03b2=1/2[cos(\u03b1+\u03b2)+cos(\u03b1-\u03b2)]
7. \u534a\u89d2\u516c\u5f0f
\u53e6\uff1a\u4e09\u89d2\u51fd\u6570\u53e3\u8bc0
\u4e09\u89d2\u77e5\u8bc6\uff0c\u81ea\u6210\u4f53\u7cfb\uff0c
\u8bb0\u5fc6\u53e3\u8bc0\uff0c\u4e00\u4e8c\u4e09\u56db\u3002
\u4e00\u4e2a\u5b9a\u4e49\uff0c\u4e09\u89d2\u51fd\u6570\uff0c
\u4e24\u79cd\u5236\u5ea6\uff0c\u89d2\u5ea6\u5f27\u5ea6\u3002
\u4e09\u5957\u516c\u5f0f\uff0c\u7262\u56fa\u8bb0\u5fc6\uff0c
\u540c\u89d2\u8bf1\u5bfc\uff0c\u52a0\u6cd5\u5b9a\u7406\u3002
\u540c\u89d2\u516c\u5f0f\uff0c\u516b\u4e2a\u4e09\u7ec4\uff0c
\u5e73\u65b9\u5173\u7cfb\uff0c\u5bfc\u6570\u5546\u6570\u3002
\u8bf1\u5bfc\u516c\u5f0f\uff0c\u4e24\u7c7b\u4e5d\u7ec4\uff0c
\u8c61\u9650\u5b9a\u53f7\uff0c\u5076\u540c\u5947\u4f59\u3002
\u4e24\u89d2\u548c\u5dee\uff0c\u6b32\u6c42\u6b63\u5f26\uff0c
\u6b63\u4f59\u4f59\u6b63\uff0c\u7b26\u53f7\u540c\u524d\u3002
\u4e24\u89d2\u548c\u5dee\uff0c\u6b32\u6c42\u4f59\u5f26\uff0c
\u4f59\u4f59\u6b63\u6b63\uff0c\u7b26\u53f7\u76f8\u53cd\u3002
\u4e24\u89d2\u76f8\u7b49\uff0c\u500d\u89d2\u516c\u5f0f\uff0c
\u9006\u5411\u53cd\u63a8\uff0c\u534a\u89d2\u6781\u9650\u3002
\u52a0\u52a0\u51cf\u51cf\uff0c\u53d8\u91cf\u66ff\u6362\uff0c
\u79ef\u5316\u548c\u5dee\uff0c\u548c\u5947\u4e92\u53d8\u3002
\u3000\u3000\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u516c\u5f0f
\u3000\u3000sin \u03b1=\u2220\u03b1\u7684\u5bf9\u8fb9 / \u659c\u8fb9
\u3000\u3000cos \u03b1=\u2220\u03b1\u7684\u90bb\u8fb9 / \u659c\u8fb9
\u3000\u3000tan \u03b1=\u2220\u03b1\u7684\u5bf9\u8fb9 / \u2220\u03b1\u7684\u90bb\u8fb9
\u3000\u3000cot \u03b1=\u2220\u03b1\u7684\u90bb\u8fb9 / \u2220\u03b1\u7684\u5bf9\u8fb9
\u3000\u3000\u500d\u89d2\u516c\u5f0f
\u3000\u3000Sin2A=2SinA?CosA
\u3000\u3000Cos2A=CosA^2-SinA^2=1-2SinA^2=2CosA^2-1
\u3000\u3000tan2A=\uff082tanA\uff09/\uff081-tanA^2\uff09
\u3000\u3000\uff08\u6ce8\uff1aSinA^2 \u662fsinA\u7684\u5e73\u65b9 sin2\uff08A\uff09 \uff09
\u3000\u3000\u4e09\u500d\u89d2\u516c\u5f0f
\u3000\u3000sin3\u03b1=4sin\u03b1•sin(\u03c0/3+\u03b1)sin(\u03c0/3-\u03b1)
\u3000\u3000cos3\u03b1=4cos\u03b1•cos(\u03c0/3+\u03b1)cos(\u03c0/3-\u03b1)
\u3000\u3000tan3a = tan a • tan(\u03c0/3+a)• tan(\u03c0/3-a)
\u3000\u3000\u4e09\u500d\u89d2\u516c\u5f0f\u63a8\u5bfc
\u3000\u3000sin3a
\u3000\u3000=sin(2a+a)
\u3000\u3000=sin2acosa+cos2asina
\u3000\u3000\u8f85\u52a9\u89d2\u516c\u5f0f
\u3000\u3000Asin\u03b1+Bcos\u03b1=(A^2+B^2)^(1/2)sin(\u03b1+t)\uff0c\u5176\u4e2d
\u3000\u3000sint=B/(A^2+B^2)^(1/2)
\u3000\u3000cost=A/(A^2+B^2)^(1/2)
\u3000\u3000tant=B/A
\u3000\u3000Asin\u03b1+Bcos\u03b1=(A^2+B^2)^(1/2)cos(\u03b1-t)\uff0ctant=A/B
\u3000\u3000\u964d\u5e42\u516c\u5f0f
\u3000\u3000sin^2(\u03b1)=(1-cos(2\u03b1))/2=versin(2\u03b1)/2
\u3000\u3000cos^2(\u03b1)=(1+cos(2\u03b1))/2=covers(2\u03b1)/2
\u3000\u3000tan^2(\u03b1)=(1-cos(2\u03b1))/(1+cos(2\u03b1))
\u3000\u3000\u63a8\u5bfc\u516c\u5f0f
\u3000\u3000tan\u03b1+cot\u03b1=2/sin2\u03b1
\u3000\u3000tan\u03b1-cot\u03b1=-2cot2\u03b1
\u3000\u30001+cos2\u03b1=2cos^2\u03b1
\u3000\u30001-cos2\u03b1=2sin^2\u03b1
\u3000\u30001+sin\u03b1=(sin\u03b1/2+cos\u03b1/2)^2
\u3000\u3000=2sina(1-sin²a)+(1-2sin²a)sina
\u3000\u3000=3sina-4sin³a
\u3000\u3000cos3a
\u3000\u3000=cos(2a+a)
\u3000\u3000=cos2acosa-sin2asina
\u3000\u3000=(2cos²a-1)cosa-2(1-sin²a)cosa
\u3000\u3000=4cos³a-3cosa
\u3000\u3000sin3a=3sina-4sin³a
\u3000\u3000=4sina(3/4-sin²a)
\u3000\u3000=4sina[(\u221a3/2)²-sin²a]
\u3000\u3000=4sina(sin²60\u00b0-sin²a)
\u3000\u3000=4sina(sin60\u00b0+sina)(sin60\u00b0-sina)
\u3000\u3000=4sina*2sin[(60+a)/2]cos[(60\u00b0-a)/2]*2sin[(60\u00b0-a)/2]cos[(60\u00b0-a)/2]
\u3000\u3000=4sinasin(60\u00b0+a)sin(60\u00b0-a)
\u3000\u3000cos3a=4cos³a-3cosa
\u3000\u3000=4cosa(cos²a-3/4)
\u3000\u3000=4cosa[cos²a-(\u221a3/2)²]
\u3000\u3000=4cosa(cos²a-cos²30\u00b0)
\u3000\u3000=4cosa(cosa+cos30\u00b0)(cosa-cos30\u00b0)
\u3000\u3000=4cosa*2cos[(a+30\u00b0)/2]cos[(a-30\u00b0)/2]*{-2sin[(a+30\u00b0)/2]sin[(a-30\u00b0)/2]}
\u3000\u3000=-4cosasin(a+30\u00b0)sin(a-30\u00b0)
\u3000\u3000=-4cosasin[90\u00b0-(60\u00b0-a)]sin[-90\u00b0+(60\u00b0+a)]
\u3000\u3000=-4cosacos(60\u00b0-a)[-cos(60\u00b0+a)]
\u3000\u3000=4cosacos(60\u00b0-a)cos(60\u00b0+a)
\u3000\u3000\u4e0a\u8ff0\u4e24\u5f0f\u76f8\u6bd4\u53ef\u5f97
\u3000\u3000tan3a=tanatan(60\u00b0-a)tan(60\u00b0+a)
\u3000\u3000\u534a\u89d2\u516c\u5f0f
\u3000\u3000tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA);
\u3000\u3000cot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA.
\u3000\u3000sin^2(a/2)=(1-cos(a))/2
\u3000\u3000cos^2(a/2)=(1+cos(a))/2
\u3000\u3000tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))
\u3000\u3000\u4e09\u89d2\u548c
\u3000\u3000sin(\u03b1+\u03b2+\u03b3)=sin\u03b1•cos\u03b2•cos\u03b3+cos\u03b1•sin\u03b2•cos\u03b3+cos\u03b1•cos\u03b2•sin\u03b3-sin\u03b1•sin\u03b2•sin\u03b3
\u3000\u3000cos(\u03b1+\u03b2+\u03b3)=cos\u03b1•cos\u03b2•cos\u03b3-cos\u03b1•sin\u03b2•sin\u03b3-sin\u03b1•cos\u03b2•sin\u03b3-sin\u03b1•sin\u03b2•cos\u03b3
\u3000\u3000tan(\u03b1+\u03b2+\u03b3)=(tan\u03b1+tan\u03b2+tan\u03b3-tan\u03b1•tan\u03b2•tan\u03b3)/(1-tan\u03b1•tan\u03b2-tan\u03b2•tan\u03b3-tan\u03b3•tan\u03b1)
\u3000\u3000\u4e24\u89d2\u548c\u5dee
\u3000\u3000cos(\u03b1+\u03b2)=cos\u03b1•cos\u03b2-sin\u03b1•sin\u03b2
\u3000\u3000cos(\u03b1-\u03b2)=cos\u03b1•cos\u03b2+sin\u03b1•sin\u03b2
\u3000\u3000sin(\u03b1\u00b1\u03b2)=sin\u03b1•cos\u03b2\u00b1cos\u03b1•sin\u03b2
\u3000\u3000tan(\u03b1+\u03b2)=(tan\u03b1+tan\u03b2)/(1-tan\u03b1•tan\u03b2)
\u3000\u3000tan(\u03b1-\u03b2)=(tan\u03b1-tan\u03b2)/(1+tan\u03b1•tan\u03b2)
\u3000\u3000\u548c\u5dee\u5316\u79ef
\u3000\u3000sin\u03b8+sin\u03c6 = 2 sin[(\u03b8+\u03c6)/2] cos[(\u03b8-\u03c6)/2]
\u3000\u3000sin\u03b8-sin\u03c6 = 2 cos[(\u03b8+\u03c6)/2] sin[(\u03b8-\u03c6)/2]
\u3000\u3000cos\u03b8+cos\u03c6 = 2 cos[(\u03b8+\u03c6)/2] cos[(\u03b8-\u03c6)/2]
\u3000\u3000cos\u03b8-cos\u03c6 = -2 sin[(\u03b8+\u03c6)/2] sin[(\u03b8-\u03c6)/2]
\u3000\u3000tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB)
\u3000\u3000tanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)
\u3000\u3000\u79ef\u5316\u548c\u5dee
\u3000\u3000sin\u03b1sin\u03b2 = [cos(\u03b1-\u03b2)-cos(\u03b1+\u03b2)] /2
\u3000\u3000cos\u03b1cos\u03b2 = [cos(\u03b1+\u03b2)+cos(\u03b1-\u03b2)]/2
\u3000\u3000sin\u03b1cos\u03b2 = [sin(\u03b1+\u03b2)+sin(\u03b1-\u03b2)]/2
\u3000\u3000cos\u03b1sin\u03b2 = [sin(\u03b1+\u03b2)-sin(\u03b1-\u03b2)]/2
\u3000\u3000\u8bf1\u5bfc\u516c\u5f0f
\u3000\u3000sin(-\u03b1) = -sin\u03b1
\u3000\u3000cos(-\u03b1) = cos\u03b1
\u3000\u3000tan (\u2014a)=-tan\u03b1
\u3000\u3000sin(\u03c0/2-\u03b1) = cos\u03b1
\u3000\u3000cos(\u03c0/2-\u03b1) = sin\u03b1
\u3000\u3000sin(\u03c0/2+\u03b1) = cos\u03b1
\u3000\u3000cos(\u03c0/2+\u03b1) = -sin\u03b1
\u3000\u3000sin(\u03c0-\u03b1) = sin\u03b1
\u3000\u3000cos(\u03c0-\u03b1) = -cos\u03b1
\u3000\u3000sin(\u03c0+\u03b1) = -sin\u03b1
\u3000\u3000cos(\u03c0+\u03b1) = -cos\u03b1
\u3000\u3000tanA= sinA/cosA
\u3000\u3000tan\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
\u3000\u3000tan\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
\u3000\u3000tan\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
\u3000\u3000tan\uff08\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
\u3000\u3000\u8bf1\u5bfc\u516c\u5f0f\u8bb0\u80cc\u8bc0\u7a8d\uff1a\u5947\u53d8\u5076\u4e0d\u53d8\uff0c\u7b26\u53f7\u770b\u8c61\u9650
\u3000\u3000\u4e07\u80fd\u516c\u5f0f
\u3000\u3000sin\u03b1=2tan(\u03b1/2\uff09/\uff3b1+tan\uff3e(\u03b1/2)\uff3d
\u3000\u3000cos\u03b1=\uff3b1-tan\uff3e(\u03b1/2)\uff3d/1+tan\uff3e(\u03b1/2)\uff3d
\u3000\u3000tan\u03b1=2tan(\u03b1/2)/\uff3b1-tan\uff3e(\u03b1/2)\uff3d
\u3000\u3000\u5176\u5b83\u516c\u5f0f
\u3000\u3000(1)(sin\u03b1)^2+(cos\u03b1)^2=1
\u3000\u3000(2)1+(tan\u03b1)^2=(sec\u03b1)^2
\u3000\u3000(3)1+(cot\u03b1)^2=(csc\u03b1)^2
\u3000\u3000\u8bc1\u660e\u4e0b\u9762\u4e24\u5f0f,\u53ea\u9700\u5c06\u4e00\u5f0f,\u5de6\u53f3\u540c\u9664(sin\u03b1)^2,\u7b2c\u4e8c\u4e2a\u9664(cos\u03b1)^2\u5373\u53ef
\u3000\u3000(4)\u5bf9\u4e8e\u4efb\u610f\u975e\u76f4\u89d2\u4e09\u89d2\u5f62,\u603b\u6709
\u3000\u3000tanA+tanB+tanC=tanAtanBtanC
\u3000\u3000\u8bc1:
\u3000\u3000A+B=\u03c0-C
\u3000\u3000tan(A+B)=tan(\u03c0-C)
\u3000\u3000(tanA+tanB)/(1-tanAtanB)=(tan\u03c0-tanC)/(1+tan\u03c0tanC)
\u3000\u3000\u6574\u7406\u53ef\u5f97
\u3000\u3000tanA+tanB+tanC=tanAtanBtanC
\u3000\u3000\u5f97\u8bc1
\u3000\u3000\u540c\u6837\u53ef\u4ee5\u5f97\u8bc1,\u5f53x+y+z=n\u03c0(n\u2208Z)\u65f6,\u8be5\u5173\u7cfb\u5f0f\u4e5f\u6210\u7acb
\u3000\u3000\u7531tanA+tanB+tanC=tanAtanBtanC\u53ef\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba
\u3000\u3000(5)cotAcotB+cotAcotC+cotBcotC=1
\u3000\u3000(6)cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2)
\u3000\u3000(7)(cosA\uff09^2+(cosB\uff09^2+(cosC\uff09^2=1-2cosAcosBcosC
\u3000\u3000(8)\uff08sinA\uff09^2+\uff08sinB\uff09^2+\uff08sinC\uff09^2=2+2cosAcosBcosC
\u3000\u3000(9)sin\u03b1+sin(\u03b1+2\u03c0/n)+sin(\u03b1+2\u03c0*2/n)+sin(\u03b1+2\u03c0*3/n)+\u2026\u2026+sin[\u03b1+2\u03c0*(n-1)/n]=0
\u3000\u3000cos\u03b1+cos(\u03b1+2\u03c0/n)+cos(\u03b1+2\u03c0*2/n)+cos(\u03b1+2\u03c0*3/n)+\u2026\u2026+cos[\u03b1+2\u03c0*(n-1)/n]=0 \u4ee5\u53ca
\u3000\u3000sin^2(\u03b1)+sin^2(\u03b1-2\u03c0/3)+sin^2(\u03b1+2\u03c0/3)=3/2
\u3000\u3000tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0
\u4e0d\u77e5\u9053\u8fd9\u6837\u80fd\u4e0d\u80fd\u6ee1\u8db3\u4f60\u7684\u8981\u6c42\uff0c\u56fe\u50cf\u771f\u7684\u4e0d\u597d\u5f04\u554a\uff0c\u4f1a\u662f\u4f1a\uff0csin \u548ccos \u7ecf\u5e38\u89c1\uff0ctan\u56fe\u50cf\u597d\u591a\u4eba\u4e0d\u77e5\u9053\u7684\uff0c\u56e0\u4e3a\u5f88\u5c11\u89c1\uff0c\u662f\u5947\u51fd\u6570\uff0c\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\uff0c\u589e\u51fd\u6570\u5728\u539f\u70b9\u4e24\u8fb9\uff0c\u662f\u4e00\u4e2a\u5b8c\u6574\u7684\u56fe\u50cf\uff0c\u8fd8\u6709\u5c31\u662f\u5468\u671f\u662f\u03c0\uff0c\uff08-\u03c0/2~\u03c0/2\uff09\u662f\u4e00\u4e2a\u5468\u671f\u3002cot\u7684\u56fe\u50cf\u662f\u51cf\u51fd\u6570\uff0c\uff080~\u03c0\uff09\u662f\u4e00\u4e2a\u5468\u671f\u3002
公式六是pi/2+a
根据角的不同选取公式，对于这两组公式，变化后函数的名称都有要变化。
因为sin(兀/2+a)=cosa就是诱导公式6
sin(兀/2-a)=cosa是诱导公式5
绛旓細鍏紡浜旀槸pi/2-a 鍏紡鍏槸pi/2+a 鏍规嵁瑙掔殑涓嶅悓閫夊彇鍏紡锛屽浜庤繖涓ょ粍鍏紡锛屽彉鍖栧悗鍑芥暟鐨勫悕绉伴兘鏈夎鍙樺寲銆
绛旓細tan(蟺-x)=-tanx 鍘熺悊锛涓夎鍑芥暟鍊间腑锛屾寮︿竴浜岃薄闄愪负姝ｏ紝浣欏鸡涓鍥涜薄闄愪负姝ｏ紝姝ｅ垏涓涓夎薄闄愪负姝ｏ紙缁堣竟锛(5)sin(蟺/2+x)=cosx cos(蟺/2+x)=-sinx tan(蟺/2+x)=-cotx (6)sin(蟺/2-x)=cosx cos(蟺/2-x)=sinx tan(蟺/2-x)=cotx (7)灞曞紑鍏紡 sin(3蟺/2+x)=sin(蟺+蟺/2+...
绛旓細cot(蟺/2-伪)=tan伪 涓夎鍑芥暟璇卞鍏紡涓 鎺ㄧ畻鍏紡锛3蟺/2卤伪涓幬辩殑涓夎鍑芥暟鍊间箣闂寸殑鍏崇郴锛歴in(3蟺/2+伪)=-cos伪cos(3蟺/2+伪)=sin伪tan(3蟺/2+伪)=-cot伪cot(3蟺/2+伪)=-tan伪sin(3蟺/2-伪)=-cos伪cos(3蟺/2-伪)=-sin伪tan(3蟺/2-伪)=cot伪cot(3蟺/2-伪)=tan伪...
绛旓細涓夎鍑芥暟璇卞鍏紡锛氫笁瑙掑嚱鏁扮殑鍩烘湰鍏紡锛1銆佸叕寮忎竴锛氫换鎰忚伪涓-伪鐨勪笁瑙掑嚱鏁板间箣闂寸殑鍏崇郴锛歴in锛堬紞伪锛=锛峴in伪cos锛堬紞伪锛=cos伪tan锛堬紞伪锛=锛峵an伪cot锛堬紞伪锛=锛峜ot伪 2銆佸叕寮忎簩锛歴in锛埾+伪锛=鈥攕in伪cos锛埾+伪锛=锛峜os伪tan锛埾+伪锛=tan伪cot锛埾+伪锛=cot伪 3銆佸叕寮忎笁锛氬埄鐢...
绛旓細1銆鍏紡涓:璁疚变负浠绘剰瑙掞紝缁堣竟鐩稿悓鐨勮鐨勫悓涓涓夎鍑芥暟鐨勫肩浉绛夛細sin(2k蟺+伪)=sin伪(k鈭圸)cos(2k蟺+伪)=cos伪(k鈭圸)tan(2k蟺+伪)=tan伪(k鈭圸)cot(2k蟺+伪)=cot伪(k鈭圸)2銆佸叕寮忎簩:璁疚变负浠绘剰瑙掞紝蟺+伪鐨勪笁瑙掑嚱鏁板间笌伪鐨勪笁瑙掑嚱鏁板间箣闂寸殑鍏崇郴锛歴in(蟺+伪)=锛峴in伪 cos(蟺+伪...
绛旓細涓夎鍑芥暟璇卞鍏紡涓锛氫换鎰忚伪涓-伪鐨勪笁瑙掑嚱鏁板间箣闂寸殑鍏崇郴锛歴in(-伪)=-sin伪 cos(-伪)=cos伪 tan(-伪)=-tan伪 cot(-伪)=-cot伪 涓夎鍑芥暟璇卞鍏紡浜岋細璁疚变负浠绘剰瑙掞紝蟺+伪鐨勪笁瑙掑嚱鏁板间笌伪鐨勪笁瑙掑嚱鏁板间箣闂寸殑鍏崇郴锛歴in(蟺+伪)=-sin伪 cos(蟺+伪)=-cos伪 tan(蟺+伪)=tan伪 cot(蟺...
绛旓細涓夎鍑芥暟鐨璇卞鍏紡鏄竴缁勭敤浜庡皢瑙掑害杞崲涓哄叾浠栧舰寮忕殑鍏紡銆傜浉鍏崇煡璇嗗涓嬶細1銆佹寮﹀嚱鏁扮殑璇卞鍏紡锛歴in锛坸+2蟺锛=sin锛坸锛夛紝sin锛坸+蟺锛=-sin锛坸锛夛紝sin锛坸+蟺/2锛=cos锛坸锛夛紝sin锛坸-蟺/2锛=-cos锛坸锛夈備綑寮﹀嚱鏁扮殑璇卞鍏紡锛歝os锛坸+2蟺锛=cos锛坸锛夛紝cos锛坸+蟺锛=-cos锛坸锛夛紝cos锛坸...
绛旓細涓夎鍑芥暟璇卞鍏紡鎺ㄥ杩囩▼ sin锛堬紞a锛夛紳锛峴ina锛宻in锛堬紞a锛夛紳sin锛0锛峚锛夛紳sin0cosa锛峴inacos0锛0锛峴ina锛濓紞sina锛沜os锛堬紞a锛夛紳cosa锛宑os锛堬紞a锛夛紳cos锛0锛峚锛夛紳cos0cosa锛媠in0sina锛漜osa锛0锛漜osa锛泂in锛埾锛2锛峚锛夛紳cosa锛宻in锛埾锛2锛峚锛夛紳sin蟺锛2cosa锛峴inacos蟺锛2锛漜osa锛0锛...
绛旓細璇卞鍏紡涓夎鍑芥暟鍩烘湰鍏紡濡備笅锛歴in锛2k蟺+伪锛=sin伪锛坘鈭圸锛塩os锛2k蟺+伪锛=cos伪锛坘鈭圸锛塼an锛2k蟺+伪锛=tan伪锛坘鈭圸锛塩ot锛2k蟺+伪锛=cot伪锛坘鈭圸锛夎瀵煎叕寮忓彛璇鈥滃鍙樺伓涓嶅彉锛岀鍙风湅璞￠檺鈥濇剰涔夛細k脳蟺/2卤a锛坘鈭圸锛夌殑涓夎鍑芥暟鍊硷細锛1锛夊綋k涓哄伓鏁版椂锛岀瓑浜幬辩殑鍚屽悕涓夎鍑芥暟鍊硷紝...
绛旓細鍏紡涓锛氬浜庝换鎰忚伪锛屾墍鏈変笌伪缁堣竟鐩稿悓鐨勮锛屽叾涓夎鍑芥暟鍊兼亽绛夛紝濡傦細sin(2k蟺+伪) = sin伪, cos(2k蟺+伪) = cos伪, tan(2k蟺+伪) = tan伪, cot(2k蟺+伪) = cot伪, 鍏朵腑k涓烘暣鏁般傚叕寮忎簩锛毾+伪鐨勪笁瑙掑嚱鏁颁笌伪鐨勭浉瀵瑰叧绯伙細sin(蟺+伪) = -sin伪, cos(蟺+伪) = -cos伪, tan...
