高中数学必修4三角函数公式的证明，两角和差，二倍角，降幂升幂，半角，辅助，积化和差，和差化积 高一必修4里所有三角函数公式及其推导过程
\u9ad8\u4e2d\u6570\u5b66\u5fc5\u4fee4\u4e09\u89d2\u51fd\u6570\u516c\u5f0f\u5927\u5168\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\uff1dsin\u03b1
cos\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcos\u03b1
tan\uff082k\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
cot\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcot\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\uff1d\uff0dsin\u03b1
cos\uff08\u03c0\uff0b\u03b1\uff09\uff1d\uff0dcos\u03b1
tan\uff08\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
cot\uff08\u03c0\uff0b\u03b1\uff09\uff1dcot\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\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
cos\uff08\uff0d\u03b1\uff09\uff1dcos\u03b1
tan\uff08\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
cot\uff08\uff0d\u03b1\uff09\uff1d\uff0dcot\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\uff0d\u03b1\uff09\uff1dsin\u03b1
cos\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcos\u03b1
tan\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
cot\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcot\u03b1
\u516c\u5f0f\u4e94\uff1a
\u5229\u7528\u516c\u5f0f\u4e00\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\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
cos\uff082\u03c0\uff0d\u03b1\uff09\uff1dcos\u03b1
tan\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
cot\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcot\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\uff0b\u03b1\uff09\uff1dcos\u03b1
cos\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dsin\u03b1
tan\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
cot\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dtan\u03b1
sin\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcos\u03b1
cos\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dsin\u03b1
tan\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
cot\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dtan\u03b1
sin\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcos\u03b1
cos\uff083\u03c0/2\uff0b\u03b1\uff09\uff1dsin\u03b1
tan\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
cot\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dtan\u03b1
sin\uff083\u03c0/2\uff0d\u03b1\uff09\uff1d\uff0dcos\u03b1
cos\uff083\u03c0/2\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
tan\uff083\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
cot\uff083\u03c0/2\uff0d\u03b1\uff09\uff1dtan\u03b1
(\u4ee5\u4e0ak\u2208Z)
\u8bf1\u5bfc\u516c\u5f0f\u8bb0\u5fc6\u53e3\u8bc0
\u203b\u89c4\u5f8b\u603b\u7ed3\u203b
\u4e0a\u9762\u8fd9\u4e9b\u8bf1\u5bfc\u516c\u5f0f\u53ef\u4ee5\u6982\u62ec\u4e3a\uff1a
\u5bf9\u4e8ek\u00b7\u03c0/2\u00b1\u03b1(k\u2208Z)\u7684\u4e2a\u4e09\u89d2\u51fd\u6570\u503c\uff0c
\u2460\u5f53k\u662f\u5076\u6570\u65f6\uff0c\u5f97\u5230\u03b1\u7684\u540c\u540d\u51fd\u6570\u503c\uff0c\u5373\u51fd\u6570\u540d\u4e0d\u6539\u53d8\uff1b
\u2461\u5f53k\u662f\u5947\u6570\u65f6\uff0c\u5f97\u5230\u03b1\u76f8\u5e94\u7684\u4f59\u51fd\u6570\u503c\uff0c\u5373sin\u2192cos;cos\u2192sin;tan\u2192cot,cot\u2192tan.
\uff08\u5947\u53d8\u5076\u4e0d\u53d8\uff09
\u7136\u540e\u5728\u524d\u9762\u52a0\u4e0a\u628a\u03b1\u770b\u6210\u9510\u89d2\u65f6\u539f\u51fd\u6570\u503c\u7684\u7b26\u53f7\u3002
\uff08\u7b26\u53f7\u770b\u8c61\u9650\uff09
\u4f8b\u5982\uff1a
sin(2\u03c0\uff0d\u03b1)\uff1dsin(4\u00b7\u03c0/2\uff0d\u03b1)\uff0ck\uff1d4\u4e3a\u5076\u6570\uff0c\u6240\u4ee5\u53d6sin\u03b1\u3002
\u5f53\u03b1\u662f\u9510\u89d2\u65f6\uff0c2\u03c0\uff0d\u03b1\u2208(270\u00b0\uff0c360\u00b0)\uff0csin(2\u03c0\uff0d\u03b1)\uff1c0\uff0c\u7b26\u53f7\u4e3a\u201c\uff0d\u201d\u3002
\u6240\u4ee5sin(2\u03c0\uff0d\u03b1)\uff1d\uff0dsin\u03b1
\u4e0a\u8ff0\u7684\u8bb0\u5fc6\u53e3\u8bc0\u662f\uff1a
\u5947\u53d8\u5076\u4e0d\u53d8\uff0c\u7b26\u53f7\u770b\u8c61\u9650\u3002
\u516c\u5f0f\u53f3\u8fb9\u7684\u7b26\u53f7\u4e3a\u628a\u03b1\u89c6\u4e3a\u9510\u89d2\u65f6\uff0c\u89d2k\u00b7360\u00b0+\u03b1\uff08k\u2208Z\uff09\uff0c-\u03b1\u3001180\u00b0\u00b1\u03b1\uff0c360\u00b0-\u03b1
\u6240\u5728\u8c61\u9650\u7684\u539f\u4e09\u89d2\u51fd\u6570\u503c\u7684\u7b26\u53f7\u53ef\u8bb0\u5fc6
\u6c34\u5e73\u8bf1\u5bfc\u540d\u4e0d\u53d8\uff1b\u7b26\u53f7\u770b\u8c61\u9650\u3002
\u5404\u79cd\u4e09\u89d2\u51fd\u6570\u5728\u56db\u4e2a\u8c61\u9650\u7684\u7b26\u53f7\u5982\u4f55\u5224\u65ad\uff0c\u4e5f\u53ef\u4ee5\u8bb0\u4f4f\u53e3\u8bc0\u201c\u4e00\u5168\u6b63\uff1b\u4e8c\u6b63\u5f26\uff1b\u4e09\u4e3a\u5207\uff1b\u56db\u4f59\u5f26\u201d\uff0e
\u8fd9\u5341\u4e8c\u5b57\u53e3\u8bc0\u7684\u610f\u601d\u5c31\u662f\u8bf4\uff1a
\u7b2c\u4e00\u8c61\u9650\u5185\u4efb\u4f55\u4e00\u4e2a\u89d2\u7684\u56db\u79cd\u4e09\u89d2\u51fd\u6570\u503c\u90fd\u662f\u201c\uff0b\u201d\uff1b
\u7b2c\u4e8c\u8c61\u9650\u5185\u53ea\u6709\u6b63\u5f26\u662f\u201c\uff0b\u201d\uff0c\u5176\u4f59\u5168\u90e8\u662f\u201c\uff0d\u201d\uff1b
\u7b2c\u4e09\u8c61\u9650\u5185\u5207\u51fd\u6570\u662f\u201c\uff0b\u201d\uff0c\u5f26\u51fd\u6570\u662f\u201c\uff0d\u201d\uff1b
\u7b2c\u56db\u8c61\u9650\u5185\u53ea\u6709\u4f59\u5f26\u662f\u201c\uff0b\u201d\uff0c\u5176\u4f59\u5168\u90e8\u662f\u201c\uff0d\u201d\uff0e
\u5176\u4ed6\u4e09\u89d2\u51fd\u6570\u77e5\u8bc6\uff1a
\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u57fa\u672c\u5173\u7cfb
\u2488\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u7684\u57fa\u672c\u5173\u7cfb\u5f0f
\u5012\u6570\u5173\u7cfb:
tan\u03b1 \u00b7cot\u03b1\uff1d1
sin\u03b1 \u00b7csc\u03b1\uff1d1
cos\u03b1 \u00b7sec\u03b1\uff1d1
\u5546\u7684\u5173\u7cfb\uff1a
sin\u03b1/cos\u03b1\uff1dtan\u03b1\uff1dsec\u03b1/csc\u03b1
cos\u03b1/sin\u03b1\uff1dcot\u03b1\uff1dcsc\u03b1/sec\u03b1
\u5e73\u65b9\u5173\u7cfb\uff1a
sin^2(\u03b1)\uff0bcos^2(\u03b1)\uff1d1
1\uff0btan^2(\u03b1)\uff1dsec^2(\u03b1)
1\uff0bcot^2(\u03b1)\uff1dcsc^2(\u03b1)
\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u5173\u7cfb\u516d\u89d2\u5f62\u8bb0\u5fc6\u6cd5
\u516d\u89d2\u5f62\u8bb0\u5fc6\u6cd5\uff1a\uff08\u53c2\u770b\u56fe\u7247\u6216\u53c2\u8003\u8d44\u6599\u94fe\u63a5\uff09
\u6784\u9020\u4ee5"\u4e0a\u5f26\u3001\u4e2d\u5207\u3001\u4e0b\u5272\uff1b\u5de6\u6b63\u3001\u53f3\u4f59\u3001\u4e2d\u95f41"\u7684\u6b63\u516d\u8fb9\u5f62\u4e3a\u6a21\u578b\u3002
\uff081\uff09\u5012\u6570\u5173\u7cfb\uff1a\u5bf9\u89d2\u7ebf\u4e0a\u4e24\u4e2a\u51fd\u6570\u4e92\u4e3a\u5012\u6570\uff1b
\uff082\uff09\u5546\u6570\u5173\u7cfb\uff1a\u516d\u8fb9\u5f62\u4efb\u610f\u4e00\u9876\u70b9\u4e0a\u7684\u51fd\u6570\u503c\u7b49\u4e8e\u4e0e\u5b83\u76f8\u90bb\u7684\u4e24\u4e2a\u9876\u70b9\u4e0a\u51fd\u6570\u503c\u7684\u4e58\u79ef\u3002
\uff08\u4e3b\u8981\u662f\u4e24\u6761\u865a\u7ebf\u4e24\u7aef\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7684\u4e58\u79ef\uff09\u3002\u7531\u6b64\uff0c\u53ef\u5f97\u5546\u6570\u5173\u7cfb\u5f0f\u3002
\uff083\uff09\u5e73\u65b9\u5173\u7cfb\uff1a\u5728\u5e26\u6709\u9634\u5f71\u7ebf\u7684\u4e09\u89d2\u5f62\u4e2d\uff0c\u4e0a\u9762\u4e24\u4e2a\u9876\u70b9\u4e0a\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7684\u5e73\u65b9\u548c\u7b49\u4e8e\u4e0b\u9762\u9876\u70b9\u4e0a\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7684\u5e73\u65b9\u3002
\u4e24\u89d2\u548c\u5dee\u516c\u5f0f
\u2489\u4e24\u89d2\u548c\u4e0e\u5dee\u7684\u4e09\u89d2\u51fd\u6570\u516c\u5f0f
sin\uff08\u03b1\uff0b\u03b2\uff09\uff1dsin\u03b1cos\u03b2\uff0bcos\u03b1sin\u03b2
sin\uff08\u03b1\uff0d\u03b2\uff09\uff1dsin\u03b1cos\u03b2\uff0dcos\u03b1sin\u03b2
cos\uff08\u03b1\uff0b\u03b2\uff09\uff1dcos\u03b1cos\u03b2\uff0dsin\u03b1sin\u03b2
cos\uff08\u03b1\uff0d\u03b2\uff09\uff1dcos\u03b1cos\u03b2\uff0bsin\u03b1sin\u03b2
tan\u03b1\uff0btan\u03b2
tan\uff08\u03b1\uff0b\u03b2\uff09\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0dtan\u03b1 \u00b7tan\u03b2
tan\u03b1\uff0dtan\u03b2
tan\uff08\u03b1\uff0d\u03b2\uff09\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0btan\u03b1 \u00b7tan\u03b2
\u500d\u89d2\u516c\u5f0f
\u248a\u4e8c\u500d\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f\uff08\u5347\u5e42\u7f29\u89d2\u516c\u5f0f\uff09
sin2\u03b1\uff1d2sin\u03b1cos\u03b1
cos2\u03b1\uff1dcos^2(\u03b1)\uff0dsin^2(\u03b1)\uff1d2cos^2(\u03b1)\uff0d1\uff1d1\uff0d2sin^2(\u03b1)
2tan\u03b1
tan2\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014
1\uff0dtan^2(\u03b1)
\u534a\u89d2\u516c\u5f0f
\u248b\u534a\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f\uff08\u964d\u5e42\u6269\u89d2\u516c\u5f0f\uff09
1\uff0dcos\u03b1
sin^2(\u03b1/2)\uff1d\u2014\u2014\u2014\u2014\u2014
2
1\uff0bcos\u03b1
cos^2(\u03b1/2)\uff1d\u2014\u2014\u2014\u2014\u2014
2
1\uff0dcos\u03b1
tan^2(\u03b1/2)\uff1d\u2014\u2014\u2014\u2014\u2014
1\uff0bcos\u03b1
\u4e07\u80fd\u516c\u5f0f
\u248c\u4e07\u80fd\u516c\u5f0f
2tan(\u03b1/2)
sin\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0btan^2(\u03b1/2)
1\uff0dtan^2(\u03b1/2)
cos\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0btan^2(\u03b1/2)
2tan(\u03b1/2)
tan\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0dtan^2(\u03b1/2)
\u4e07\u80fd\u516c\u5f0f\u63a8\u5bfc
\u9644\u63a8\u5bfc\uff1a
sin2\u03b1=2sin\u03b1cos\u03b1=2sin\u03b1cos\u03b1/(cos^2(\u03b1)+sin^2(\u03b1))......*\uff0c
\uff08\u56e0\u4e3acos^2(\u03b1)+sin^2(\u03b1)=1\uff09
\u518d\u628a*\u5206\u5f0f\u4e0a\u4e0b\u540c\u9664cos^2(\u03b1)\uff0c\u53ef\u5f97sin2\u03b1\uff1dtan2\u03b1/(1\uff0btan^2(\u03b1))
\u7136\u540e\u7528\u03b1/2\u4ee3\u66ff\u03b1\u5373\u53ef\u3002
\u540c\u7406\u53ef\u63a8\u5bfc\u4f59\u5f26\u7684\u4e07\u80fd\u516c\u5f0f\u3002\u6b63\u5207\u7684\u4e07\u80fd\u516c\u5f0f\u53ef\u901a\u8fc7\u6b63\u5f26\u6bd4\u4f59\u5f26\u5f97\u5230\u3002
\u4e09\u500d\u89d2\u516c\u5f0f
\u248d\u4e09\u500d\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f
sin3\u03b1\uff1d3sin\u03b1\uff0d4sin^3(\u03b1)
cos3\u03b1\uff1d4cos^3(\u03b1)\uff0d3cos\u03b1
3tan\u03b1\uff0dtan^3(\u03b1)
tan3\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0d3tan^2(\u03b1)
\u4e09\u500d\u89d2\u516c\u5f0f\u63a8\u5bfc
\u9644\u63a8\u5bfc\uff1a
tan3\u03b1\uff1dsin3\u03b1/cos3\u03b1
\uff1d(sin2\u03b1cos\u03b1\uff0bcos2\u03b1sin\u03b1)/(cos2\u03b1cos\u03b1-sin2\u03b1sin\u03b1)
\uff1d(2sin\u03b1cos^2(\u03b1)\uff0bcos^2(\u03b1)sin\u03b1\uff0dsin^3(\u03b1))/(cos^3(\u03b1)\uff0dcos\u03b1sin^2(\u03b1)\uff0d2sin^2(\u03b1)cos\u03b1)
\u4e0a\u4e0b\u540c\u9664\u4ee5cos^3(\u03b1)\uff0c\u5f97\uff1a
tan3\u03b1\uff1d(3tan\u03b1\uff0dtan^3(\u03b1))/(1-3tan^2(\u03b1))
sin3\u03b1\uff1dsin(2\u03b1\uff0b\u03b1)\uff1dsin2\u03b1cos\u03b1\uff0bcos2\u03b1sin\u03b1
\uff1d2sin\u03b1cos^2(\u03b1)\uff0b(1\uff0d2sin^2(\u03b1))sin\u03b1
\uff1d2sin\u03b1\uff0d2sin^3(\u03b1)\uff0bsin\u03b1\uff0d2sin^2(\u03b1)
\uff1d3sin\u03b1\uff0d4sin^3(\u03b1)
cos3\u03b1\uff1dcos(2\u03b1\uff0b\u03b1)\uff1dcos2\u03b1cos\u03b1\uff0dsin2\u03b1sin\u03b1
\uff1d(2cos^2(\u03b1)\uff0d1)cos\u03b1\uff0d2cos\u03b1sin^2(\u03b1)
\uff1d2cos^3(\u03b1)\uff0dcos\u03b1\uff0b(2cos\u03b1\uff0d2cos^3(\u03b1))
\uff1d4cos^3(\u03b1)\uff0d3cos\u03b1
\u5373
sin3\u03b1\uff1d3sin\u03b1\uff0d4sin^3(\u03b1)
cos3\u03b1\uff1d4cos^3(\u03b1)\uff0d3cos\u03b1
\u4e09\u500d\u89d2\u516c\u5f0f\u8054\u60f3\u8bb0\u5fc6
\u8bb0\u5fc6\u65b9\u6cd5\uff1a\u8c10\u97f3\u3001\u8054\u60f3
\u6b63\u5f26\u4e09\u500d\u89d2\uff1a3\u5143 \u51cf 4\u51433\u89d2\uff08\u6b20\u503a\u4e86(\u88ab\u51cf\u6210\u8d1f\u6570)\uff0c\u6240\u4ee5\u8981\u201c\u6323\u94b1\u201d(\u97f3\u4f3c\u201c\u6b63\u5f26\u201d)\uff09
\u4f59\u5f26\u4e09\u500d\u89d2\uff1a4\u51433\u89d2 \u51cf 3\u5143\uff08\u51cf\u5b8c\u4e4b\u540e\u8fd8\u6709\u201c\u4f59\u201d\uff09
\u2606\u2606\u6ce8\u610f\u51fd\u6570\u540d\uff0c\u5373\u6b63\u5f26\u7684\u4e09\u500d\u89d2\u90fd\u7528\u6b63\u5f26\u8868\u793a\uff0c\u4f59\u5f26\u7684\u4e09\u500d\u89d2\u90fd\u7528\u4f59\u5f26\u8868\u793a\u3002
\u548c\u5dee\u5316\u79ef\u516c\u5f0f
\u248e\u4e09\u89d2\u51fd\u6570\u7684\u548c\u5dee\u5316\u79ef\u516c\u5f0f
\u03b1\uff0b\u03b2 \u03b1\uff0d\u03b2
sin\u03b1\uff0bsin\u03b2\uff1d2sin\u2014----\u00b7cos\u2014---
2 2
\u03b1\uff0b\u03b2 \u03b1\uff0d\u03b2
sin\u03b1\uff0dsin\u03b2\uff1d2cos\u2014----\u00b7sin\u2014----
2 2
\u03b1\uff0b\u03b2 \u03b1\uff0d\u03b2
cos\u03b1\uff0bcos\u03b2\uff1d2cos\u2014-----\u00b7cos\u2014-----
2 2
\u03b1\uff0b\u03b2 \u03b1\uff0d\u03b2
cos\u03b1\uff0dcos\u03b2\uff1d\uff0d2sin\u2014-----\u00b7sin\u2014-----
2 2
\u79ef\u5316\u548c\u5dee\u516c\u5f0f
\u248f\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5316\u548c\u5dee\u516c\u5f0f
sin\u03b1 \u00b7cos\u03b2\uff1d0.5[sin\uff08\u03b1\uff0b\u03b2\uff09\uff0bsin\uff08\u03b1\uff0d\u03b2\uff09]
cos\u03b1 \u00b7sin\u03b2\uff1d0.5[sin\uff08\u03b1\uff0b\u03b2\uff09\uff0dsin\uff08\u03b1\uff0d\u03b2\uff09]
cos\u03b1 \u00b7cos\u03b2\uff1d0.5[cos\uff08\u03b1\uff0b\u03b2\uff09\uff0bcos\uff08\u03b1\uff0d\u03b2\uff09]
sin\u03b1 \u00b7sin\u03b2\uff1d\uff0d 0.5[cos\uff08\u03b1\uff0b\u03b2\uff09\uff0dcos\uff08\u03b1\uff0d\u03b2\uff09]
\u548c\u5dee\u5316\u79ef\u516c\u5f0f\u63a8\u5bfc
\u9644\u63a8\u5bfc\uff1a
\u9996\u5148,\u6211\u4eec\u77e5\u9053sin(a+b)=sina*cosb+cosa*sinb,sin(a-b)=sina*cosb-cosa*sinb
\u6211\u4eec\u628a\u4e24\u5f0f\u76f8\u52a0\u5c31\u5f97\u5230sin(a+b)+sin(a-b)=2sina*cosb
\u6240\u4ee5,sina*cosb=(sin(a+b)+sin(a-b))/2
\u540c\u7406,\u82e5\u628a\u4e24\u5f0f\u76f8\u51cf,\u5c31\u5f97\u5230cosa*sinb=(sin(a+b)-sin(a-b))/2
\u540c\u6837\u7684,\u6211\u4eec\u8fd8\u77e5\u9053cos(a+b)=cosa*cosb-sina*sinb,cos(a-b)=cosa*cosb+sina*sinb
\u6240\u4ee5,\u628a\u4e24\u5f0f\u76f8\u52a0,\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230cos(a+b)+cos(a-b)=2cosa*cosb
\u6240\u4ee5\u6211\u4eec\u5c31\u5f97\u5230,cosa*cosb=(cos(a+b)+cos(a-b))/2
\u540c\u7406,\u4e24\u5f0f\u76f8\u51cf\u6211\u4eec\u5c31\u5f97\u5230sina*sinb=-(cos(a+b)-cos(a-b))/2
\u8fd9\u6837,\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u79ef\u5316\u548c\u5dee\u7684\u56db\u4e2a\u516c\u5f0f:
sina*cosb=(sin(a+b)+sin(a-b))/2
cosa*sinb=(sin(a+b)-sin(a-b))/2
cosa*cosb=(cos(a+b)+cos(a-b))/2
sina*sinb=-(cos(a+b)-cos(a-b))/2
\u597d,\u6709\u4e86\u79ef\u5316\u548c\u5dee\u7684\u56db\u4e2a\u516c\u5f0f\u4ee5\u540e,\u6211\u4eec\u53ea\u9700\u4e00\u4e2a\u53d8\u5f62,\u5c31\u53ef\u4ee5\u5f97\u5230\u548c\u5dee\u5316\u79ef\u7684\u56db\u4e2a\u516c\u5f0f.
\u6211\u4eec\u628a\u4e0a\u8ff0\u56db\u4e2a\u516c\u5f0f\u4e2d\u7684a+b\u8bbe\u4e3ax,a-b\u8bbe\u4e3ay,\u90a3\u4e48a=(x+y)/2,b=(x-y)/2
\u628aa,b\u5206\u522b\u7528x,y\u8868\u793a\u5c31\u53ef\u4ee5\u5f97\u5230\u548c\u5dee\u5316\u79ef\u7684\u56db\u4e2a\u516c\u5f0f:
sinx+siny=2sin((x+y)/2)*cos((x-y)/2)
sinx-siny=2cos((x+y)/2)*sin((x-y)/2)
cosx+cosy=2cos((x+y)/2)*cos((x-y)/2)
cosx-cosy=-2sin((x+y)/2)*sin((x-y)/2)
\u5411\u91cf\u7684\u8fd0\u7b97
\u52a0\u6cd5\u8fd0\u7b97
AB\uff0bBC\uff1dAC\uff0c\u8fd9\u79cd\u8ba1\u7b97\u6cd5\u5219\u53eb\u505a\u5411\u91cf\u52a0\u6cd5\u7684\u4e09\u89d2\u5f62\u6cd5\u5219\u3002
\u5df2\u77e5\u4e24\u4e2a\u4ece\u540c\u4e00\u70b9O\u51fa\u53d1\u7684\u4e24\u4e2a\u5411\u91cfOA\u3001OB\uff0c\u4ee5OA\u3001OB\u4e3a\u90bb\u8fb9\u4f5c\u5e73\u884c\u56db\u8fb9\u5f62OACB\uff0c\u5219\u4ee5O\u4e3a\u8d77\u70b9\u7684\u5bf9\u89d2\u7ebfOC\u5c31\u662f\u5411\u91cfOA\u3001OB\u7684\u548c\uff0c\u8fd9\u79cd\u8ba1\u7b97\u6cd5\u5219\u53eb\u505a\u5411\u91cf\u52a0\u6cd5\u7684\u5e73\u884c\u56db\u8fb9\u5f62\u6cd5\u5219\u3002
\u5bf9\u4e8e\u96f6\u5411\u91cf\u548c\u4efb\u610f\u5411\u91cfa\uff0c\u6709\uff1a0\uff0ba\uff1da\uff0b0\uff1da\u3002
|a\uff0bb|\u2264|a|\uff0b|b|\u3002
\u5411\u91cf\u7684\u52a0\u6cd5\u6ee1\u8db3\u6240\u6709\u7684\u52a0\u6cd5\u8fd0\u7b97\u5b9a\u5f8b\u3002
\u51cf\u6cd5\u8fd0\u7b97
\u4e0ea\u957f\u5ea6\u76f8\u7b49\uff0c\u65b9\u5411\u76f8\u53cd\u7684\u5411\u91cf\uff0c\u53eb\u505aa\u7684\u76f8\u53cd\u5411\u91cf\uff0c\uff0d(\uff0da)\uff1da\uff0c\u96f6\u5411\u91cf\u7684\u76f8\u53cd\u5411\u91cf\u4ecd\u7136\u662f\u96f6\u5411\u91cf\u3002
\uff081\uff09a\uff0b(\uff0da)\uff1d(\uff0da)\uff0ba\uff1d0\uff082\uff09a\uff0db\uff1da\uff0b(\uff0db)\u3002
\u6570\u4e58\u8fd0\u7b97
\u5b9e\u6570\u03bb\u4e0e\u5411\u91cfa\u7684\u79ef\u662f\u4e00\u4e2a\u5411\u91cf\uff0c\u8fd9\u79cd\u8fd0\u7b97\u53eb\u505a\u5411\u91cf\u7684\u6570\u4e58\uff0c\u8bb0\u4f5c\u03bba\uff0c|\u03bba|\uff1d|\u03bb||a|\uff0c\u5f53\u03bb > 0\u65f6\uff0c\u03bba\u7684\u65b9\u5411\u548ca\u7684\u65b9\u5411\u76f8\u540c\uff0c\u5f53\u03bb < 0\u65f6\uff0c\u03bba\u7684\u65b9\u5411\u548ca\u7684\u65b9\u5411\u76f8\u53cd\uff0c\u5f53\u03bb = 0\u65f6\uff0c\u03bba = 0\u3002
\u8bbe\u03bb\u3001\u03bc\u662f\u5b9e\u6570\uff0c\u90a3\u4e48\uff1a\uff081\uff09(\u03bb\u03bc)a = \u03bb(\u03bca)\uff082\uff09(\u03bb + \u03bc)a = \u03bba + \u03bca\uff083\uff09\u03bb(a \u00b1 b) = \u03bba \u00b1 \u03bbb\uff084\uff09(\uff0d\u03bb)a =\uff0d(\u03bba) = \u03bb(\uff0da)\u3002
\u63a8\u5bfc\u516c\u5f0f\uff1a(a+b+c)/(sinA+sinB+sinC)=2R(\u5176\u4e2d\uff0cR\u4e3a\u5916\u63a5\u5706\u534a\u5f84)
\u3000\u3000\u7531\u6b63\u5f26\u5b9a\u7406\u6709
\u3000\u3000a/sinA=b/sinB=c/sinC=2R
\u3000\u3000\u6240\u4ee5
\u3000\u3000a=2R*sinA
\u3000\u3000b=2R*sinB
\u3000\u3000c=2R*sinC
\u3000\u3000\u52a0\u8d77\u6765a+b+c=2R*(sinA+sinB+sinC)\u5e26\u5165
\u3000\u3000(a+b+c)/(sinA+sinB+sinC)=2R*(sinA+sinB+sinC)/(sinA+sinB+sinC)=2R
\u4e24\u89d2\u548c\u516c\u5f0f
\u3000\u3000sin(A+B)=sinAcosB+cosAsinB
\u3000\u3000sin(A-B)=sinAcosB-cosAsinB
\u3000\u3000cos(A+B)=cosAcosB-sinAsinB
\u3000\u3000cos(A-B)=cosAcosB+sinAsinB
\u3000\u3000tan(A+B)=(tanA+tanB)/(1-tanAtanB)
\u3000\u3000tan(A-B)=(tanA-tanB)/(1+tanAtanB)
\u3000\u3000cot(A+B)=(cotAcotB-1)/(cotB+cotA)
\u3000\u3000cot(A-B)=(cotAcotB+1)/(cotB-cotA)
\u3000\u3000\u500d\u89d2\u516c\u5f0f
\u3000\u3000Sin2A=2SinA?CosA
\u5bf9\u6570\u7684\u6027\u8d28\u53ca\u63a8\u5bfc
\u3000\u3000\u7528^\u8868\u793a\u4e58\u65b9\uff0c\u7528log(a)(b)\u8868\u793a\u4ee5a\u4e3a\u5e95\uff0cb\u7684\u5bf9\u6570
\u3000\u3000*\u8868\u793a\u4e58\u53f7\uff0c/\u8868\u793a\u9664\u53f7
\u3000\u3000\u5b9a\u4e49\u5f0f\uff1a
\u3000\u3000\u82e5a^n=b(a>0\u4e14a\u22601)
\u3000\u3000\u5219n=log(a)(b)
\u3000\u3000\u57fa\u672c\u6027\u8d28\uff1a
\u3000\u30001.a^(log(a)(b))=b
\u3000\u30002.log(a)(MN)=log(a)(M)+log(a)(N);
\u3000\u30003.log(a)(M/N)=log(a)(M)-log(a)(N);
\u3000\u30004.log(a)(M^n)=nlog(a)(M)
\u3000\u3000\u63a8\u5bfc
\u3000\u30001.\u8fd9\u4e2a\u5c31\u4e0d\u7528\u63a8\u4e86\u5427\uff0c\u76f4\u63a5\u7531\u5b9a\u4e49\u5f0f\u53ef\u5f97(\u628a\u5b9a\u4e49\u5f0f\u4e2d\u7684[n=log(a)(b)]\u5e26\u5165a^n=b)
\u3000\u30002.
\u3000\u3000MN=M*N
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d281(\u6362\u6389M\u548cN)
\u3000\u3000a^[log(a)(MN)]=a^[log(a)(M)]*a^[log(a)(N)]
\u3000\u3000\u7531\u6307\u6570\u7684\u6027\u8d28
\u3000\u3000a^[log(a)(MN)]=a^{[log(a)(M)]+[log(a)(N)]}
\u3000\u3000\u53c8\u56e0\u4e3a\u6307\u6570\u51fd\u6570\u662f\u5355\u8c03\u51fd\u6570\uff0c\u6240\u4ee5
\u3000\u3000log(a)(MN)=log(a)(M)+log(a)(N)
\u3000\u30003.\u4e0e2\u7c7b\u4f3c\u5904\u7406
\u3000\u3000MN=M/N
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d281(\u6362\u6389M\u548cN)
\u3000\u3000a^[log(a)(M/N)]=a^[log(a)(M)]/a^[log(a)(N)]
\u3000\u3000\u7531\u6307\u6570\u7684\u6027\u8d28
\u3000\u3000a^[log(a)(M/N)]=a^{[log(a)(M)]-[log(a)(N)]}
\u3000\u3000\u53c8\u56e0\u4e3a\u6307\u6570\u51fd\u6570\u662f\u5355\u8c03\u51fd\u6570\uff0c\u6240\u4ee5
\u3000\u3000log(a)(M/N)=log(a)(M)-log(a)(N)
\u3000\u30004.\u4e0e2\u7c7b\u4f3c\u5904\u7406
\u3000\u3000M^n=M^n
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d281(\u6362\u6389M)
\u3000\u3000a^[log(a)(M^n)]={a^[log(a)(M)]}^n
\u3000\u3000\u7531\u6307\u6570\u7684\u6027\u8d28
\u3000\u3000a^[log(a)(M^n)]=a^{[log(a)(M)]*n}
\u3000\u3000\u53c8\u56e0\u4e3a\u6307\u6570\u51fd\u6570\u662f\u5355\u8c03\u51fd\u6570\uff0c\u6240\u4ee5
\u3000\u3000log(a)(M^n)=nlog(a)(M)
\u3000\u3000\u5176\u4ed6\u6027\u8d28\uff1a
\u3000\u3000\u6027\u8d28\u4e00\uff1a\u6362\u5e95\u516c\u5f0f
\u3000\u3000log(a)(N)=log(b)(N)/log(b)(a)
\u3000\u3000\u63a8\u5bfc\u5982\u4e0b
\u3000\u3000N=a^[log(a)(N)]
\u3000\u3000a=b^[log(b)(a)]
\u3000\u3000\u7efc\u5408\u4e24\u5f0f\u53ef\u5f97
\u3000\u3000N={b^[log(b)(a)]}^[log(a)(N)]=b^{[log(a)(N)]*[log(b)(a)]}
\u3000\u3000\u53c8\u56e0\u4e3aN=b^[log(b)(N)]
\u3000\u3000\u6240\u4ee5
\u3000\u3000b^[log(b)(N)]=b^{[log(a)(N)]*[log(b)(a)]}
\u3000\u3000\u6240\u4ee5
\u3000\u3000log(b)(N)=[log(a)(N)]*[log(b)(a)]{\u8fd9\u6b65\u4e0d\u660e\u767d\u6216\u6709\u7591\u95ee\u770b\u4e0a\u9762\u7684}
\u3000\u3000\u6240\u4ee5log(a)(N)=log(b)(N)/log(b)(a)
\u3000\u3000\u6027\u8d28\u4e8c\uff1a\uff08\u4e0d\u77e5\u9053\u4ec0\u4e48\u540d\u5b57\uff09
\u3000\u3000log(a^n)(b^m)=m/n*[log(a)(b)]
\u3000\u3000\u63a8\u5bfc\u5982\u4e0b
\u3000\u3000\u7531\u6362\u5e95\u516c\u5f0f[lnx\u662flog(e)(x),e\u79f0\u4f5c\u81ea\u7136\u5bf9\u6570\u7684\u5e95]
\u3000\u3000log(a^n)(b^m)=ln(a^n)/ln(b^n)
\u3000\u3000\u7531\u57fa\u672c\u6027\u8d284\u53ef\u5f97
\u3000\u3000log(a^n)(b^m)=[n*ln(a)]/[m*ln(b)]=(m/n)*{[ln(a)]/[ln(b)]}
\u3000\u3000\u518d\u7531\u6362\u5e95\u516c\u5f0f
\u3000\u3000log(a^n)(b^m)=m/n*[log(a)(b)]
\u3000\u3000--------------------------------------------\uff08\u6027\u8d28\u53ca\u63a8\u5bfc\u5b8c\uff09
\u3000\u3000\u516c\u5f0f\u4e09:
\u3000\u3000log(a)(b)=1/log(b)(a)
\u3000\u3000\u8bc1\u660e\u5982\u4e0b:
\u3000\u3000\u7531\u6362\u5e95\u516c\u5f0flog(a)(b)=log(b)(b)/log(b)(a)----\u53d6\u4ee5b\u4e3a\u5e95\u7684\u5bf9\u6570,log(b)(b)=1
\u3000\u3000=1/log(b)(a)
\u3000\u3000\u8fd8\u53ef\u53d8\u5f62\u5f97:
\u3000\u3000log(a)(b)*log(b)(a)=1
\u5e73\u65b9\u5173\u7cfb\uff1a
\u3000\u3000sin^2(\u03b1)+cos^2(\u03b1)=1
\u3000\u3000tan^2(\u03b1)+1=sec^2(\u03b1)
\u3000\u3000cot^2(\u03b1)+1=csc^2(\u03b1)
\u3000\u3000•\u5546\u7684\u5173\u7cfb\uff1a
\u3000\u3000tan\u03b1=sin\u03b1/cos\u03b1cot\u03b1=cos\u03b1/sin\u03b1
\u3000\u3000•\u5012\u6570\u5173\u7cfb\uff1a
\u3000\u3000tan\u03b1•cot\u03b1=1
\u3000\u3000sin\u03b1•csc\u03b1=1
\u3000\u3000cos\u03b1•sec\u03b1=1
\u4e07\u80fd\u516c\u5f0f\uff1a
\u3000\u3000sin\u03b1=2tan(\u03b1/2)/[1+tan^2(\u03b1/2)]
\u3000\u3000cos\u03b1=[1-tan^2(\u03b1/2)]/[1+tan^2(\u03b1/2)]
\u3000\u3000tan\u03b1=2tan(\u03b1/2)/[1-tan^2(\u03b1/2)]
\u5e38\u7528\u7684\u8bf1\u5bfc\u516c\u5f0f\u6709\u4ee5\u4e0b\u51e0\u7ec4\uff1a
\u3000\u3000\u516c\u5f0f\u4e00\uff1a
\u3000\u3000\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
\u3000\u3000sin\uff082k\u03c0\uff0b\u03b1\uff09\uff1dsin\u03b1
\u3000\u3000cos\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcos\u03b1
\u3000\u3000tan\uff082k\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
\u3000\u3000cot\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcot\u03b1
\u3000\u3000\u516c\u5f0f\u4e8c\uff1a
\u3000\u3000\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
\u3000\u3000sin\uff08\u03c0\uff0b\u03b1\uff09\uff1d\uff0dsin\u03b1
\u3000\u3000cos\uff08\u03c0\uff0b\u03b1\uff09\uff1d\uff0dcos\u03b1
\u3000\u3000tan\uff08\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
\u3000\u3000cot\uff08\u03c0\uff0b\u03b1\uff09\uff1dcot\u03b1
\u3000\u3000\u516c\u5f0f\u4e09\uff1a
\u3000\u3000\u4efb\u610f\u89d2\u03b1\u4e0e-\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
\u3000\u3000sin\uff08\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
\u3000\u3000cos\uff08\uff0d\u03b1\uff09\uff1dcos\u03b1
\u3000\u3000tan\uff08\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
\u3000\u3000cot\uff08\uff0d\u03b1\uff09\uff1d\uff0dcot\u03b1
\u3000\u3000\u516c\u5f0f\u56db\uff1a
\u3000\u3000\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
\u3000\u3000sin\uff08\u03c0\uff0d\u03b1\uff09\uff1dsin\u03b1
\u3000\u3000cos\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcos\u03b1
\u3000\u3000tan\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
\u3000\u3000cot\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcot\u03b1
\u3000\u3000\u516c\u5f0f\u4e94\uff1a
\u3000\u3000\u5229\u7528\u516c\u5f0f\u4e00\u548c\u516c\u5f0f\u4e09\u53ef\u4ee5\u5f97\u52302\u03c0-\u03b1\u4e0e\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
\u3000\u3000sin\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
\u3000\u3000cos\uff082\u03c0\uff0d\u03b1\uff09\uff1dcos\u03b1
\u3000\u3000tan\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
\u3000\u3000cot\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcot\u03b1
\u3000\u3000\u516c\u5f0f\u516d\uff1a
\u3000\u3000\u03c0/2\u00b1\u03b1\u53ca3\u03c0/2\u00b1\u03b1\u4e0e\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a
\u3000\u3000sin\uff08\u03c0/2\uff0b\u03b1\uff09\uff1dcos\u03b1
\u3000\u3000cos\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dsin\u03b1
\u3000\u3000tan\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
\u3000\u3000cot\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dtan\u03b1
\u3000\u3000sin\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcos\u03b1
\u3000\u3000cos\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dsin\u03b1
\u3000\u3000tan\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
\u3000\u3000cot\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dtan\u03b1
\u3000\u3000sin\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcos\u03b1
\u3000\u3000cos\uff083\u03c0/2\uff0b\u03b1\uff09\uff1dsin\u03b1
\u3000\u3000tan\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
\u3000\u3000cot\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dtan\u03b1
\u3000\u3000sin\uff083\u03c0/2\uff0d\u03b1\uff09\uff1d\uff0dcos\u03b1
\u3000\u3000cos\uff083\u03c0/2\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
\u3000\u3000tan\uff083\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
\u3000\u3000cot\uff083\u03c0/2\uff0d\u03b1\uff09\uff1dtan\u03b1
\u3000\u3000(\u4ee5\u4e0ak\u2208Z)
\u3000\u3000\u4e00\u822c\u7684\u6700\u5e38\u7528\u516c\u5f0f\u6709:
\u3000\u3000Sin(A+B)=SinA*CosB+SinB*CosA
\u3000\u3000Sin(A-B)=SinA*CosB-SinB*CosA
\u3000\u3000Cos(A+B)=CosA*CosB-SinA*SinB
\u3000\u3000Cos(A-B)=CosA*CosB+SinA*SinB
\u3000\u3000Tan(A+B)=(TanA+TanB)/(1-TanA*TanB)
\u3000\u3000Tan(A-B)=(TanA-TanB)/(1+TanA*TanB)
\u3000\u3000\u5e73\u65b9\u5173\u7cfb\uff1a
\u3000\u3000sin^2(\u03b1)+cos^2(\u03b1)=1
\u3000\u3000tan^2(\u03b1)+1=sec^2(\u03b1)
\u3000\u3000cot^2(\u03b1)+1=csc^2(\u03b1)
\u3000\u3000•\u79ef\u7684\u5173\u7cfb\uff1a
\u3000\u3000sin\u03b1=tan\u03b1*cos\u03b1
\u3000\u3000cos\u03b1=cot\u03b1*sin\u03b1
\u3000\u3000tan\u03b1=sin\u03b1*sec\u03b1
\u3000\u3000cot\u03b1=cos\u03b1*csc\u03b1
\u3000\u3000sec\u03b1=tan\u03b1*csc\u03b1
\u3000\u3000csc\u03b1=sec\u03b1*cot\u03b1
\u3000\u3000•\u5012\u6570\u5173\u7cfb\uff1a
\u3000\u3000tan\u03b1•cot\u03b1=1
\u3000\u3000sin\u03b1•csc\u03b1=1
\u3000\u3000cos\u03b1•sec\u03b1=1
\u3000\u3000\u76f4\u89d2\u4e09\u89d2\u5f62ABC\u4e2d,
\u3000\u3000\u89d2A\u7684\u6b63\u5f26\u503c\u5c31\u7b49\u4e8e\u89d2A\u7684\u5bf9\u8fb9\u6bd4\u659c\u8fb9,
\u3000\u3000\u4f59\u5f26\u7b49\u4e8e\u89d2A\u7684\u90bb\u8fb9\u6bd4\u659c\u8fb9
\u3000\u3000\u6b63\u5207\u7b49\u4e8e\u5bf9\u8fb9\u6bd4\u90bb\u8fb9,
\u3000\u3000\u4e09\u89d2\u51fd\u6570\u6052\u7b49\u53d8\u5f62\u516c\u5f0f
\u3000\u3000•\u4e24\u89d2\u548c\u4e0e\u5dee\u7684\u4e09\u89d2\u51fd\u6570\uff1a
\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•\u8f85\u52a9\u89d2\u516c\u5f0f\uff1a
\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\u3000•\u500d\u89d2\u516c\u5f0f\uff1a
\u3000\u3000sin(2\u03b1)=2sin\u03b1•cos\u03b1=2/(tan\u03b1+cot\u03b1)
\u3000\u3000cos(2\u03b1)=cos^2(\u03b1)-sin^2(\u03b1)=2cos^2(\u03b1)-1=1-2sin^2(\u03b1)
\u3000\u3000tan(2\u03b1)=2tan\u03b1/[1-tan^2(\u03b1)]
\u3000\u3000•\u4e09\u500d\u89d2\u516c\u5f0f\uff1a
\u3000\u3000sin(3\u03b1)=3sin\u03b1-4sin^3(\u03b1)
\u3000\u3000cos(3\u03b1)=4cos^3(\u03b1)-3cos\u03b1
\u3000\u3000•\u534a\u89d2\u516c\u5f0f\uff1a
\u3000\u3000sin(\u03b1/2)=\u00b1\u221a((1-cos\u03b1)/2)
\u3000\u3000cos(\u03b1/2)=\u00b1\u221a((1+cos\u03b1)/2)
\u3000\u3000tan(\u03b1/2)=\u00b1\u221a((1-cos\u03b1)/(1+cos\u03b1))=sin\u03b1/(1+cos\u03b1)=(1-cos\u03b1)/sin\u03b1
\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=vercos(2\u03b1)/2
\u3000\u3000tan^2(\u03b1)=(1-cos(2\u03b1))/(1+cos(2\u03b1))
\u3000\u3000•\u4e07\u80fd\u516c\u5f0f\uff1a
\u3000\u3000sin\u03b1=2tan(\u03b1/2)/[1+tan^2(\u03b1/2)]
\u3000\u3000cos\u03b1=[1-tan^2(\u03b1/2)]/[1+tan^2(\u03b1/2)]
\u3000\u3000tan\u03b1=2tan(\u03b1/2)/[1-tan^2(\u03b1/2)]
\u3000\u3000•\u79ef\u5316\u548c\u5dee\u516c\u5f0f\uff1a
\u3000\u3000sin\u03b1•cos\u03b2=(1/2)[sin(\u03b1+\u03b2)+sin(\u03b1-\u03b2)]
\u3000\u3000cos\u03b1•sin\u03b2=(1/2)[sin(\u03b1+\u03b2)-sin(\u03b1-\u03b2)]
\u3000\u3000cos\u03b1•cos\u03b2=(1/2)[cos(\u03b1+\u03b2)+cos(\u03b1-\u03b2)]
\u3000\u3000sin\u03b1•sin\u03b2=-(1/2)[cos(\u03b1+\u03b2)-cos(\u03b1-\u03b2)]
\u3000\u3000•\u548c\u5dee\u5316\u79ef\u516c\u5f0f\uff1a
\u3000\u3000sin\u03b1+sin\u03b2=2sin[(\u03b1+\u03b2)/2]cos[(\u03b1-\u03b2)/2]
\u3000\u3000sin\u03b1-sin\u03b2=2cos[(\u03b1+\u03b2)/2]sin[(\u03b1-\u03b2)/2]
\u3000\u3000cos\u03b1+cos\u03b2=2cos[(\u03b1+\u03b2)/2]cos[(\u03b1-\u03b2)/2]
\u3000\u3000cos\u03b1-cos\u03b2=-2sin[(\u03b1+\u03b2)/2]sin[(\u03b1-\u03b2)/2]
\u3000\u3000•\u5176\u4ed6\uff1a
\u3000\u3000sin\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
sin(a+b)=sinacosb+cosasinb
sin(a-b)=sinacosb-sinbcosa
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)
倍角公式
tan2a=2tana/[1-(tana)^2]
cos2a=(cosa)^2-(sina)^2=2(cosa)^2 -1=1-2(sina)^2
sin2a=2sina*cosa
半角公式
sin(a/2)=√((1-cosa)/2) sin(a/2)=-√((1-cosa)/2)
cos(a/2)=√((1+cosa)/2) cos(a/2)=-√((1+cosa)/2)
tan(a/2)=√((1-cosa)/((1+cosa)) tan(a/2)=-√((1-cosa)/((1+cosa))
cot(a/2)=√((1+cosa)/((1-cosa)) cot(a/2)=-√((1+cosa)/((1-cosa))
tan(a/2)=(1-cosa)/sina=sina/(1+cosa)
和差化积
2sinacosb=sin(a+b)+sin(a-b)
2cosasinb=sin(a+b)-sin(a-b) )
2cosacosb=cos(a+b)-sin(a-b)
-2sinasinb=cos(a+b)-cos(a-b)
sina+sinb=2sin((a+b)/2)cos((a-b)/2
cosa+cosb=2cos((a+b)/2)sin((a-b)/2)
tana+tanb=sin(a+b)/cosacosb
积化和差公式
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)]
诱导公式
sin(-a)=-sin(a)
cos(-a)=cos(a)
sin(pi/2-a)=cos(a)
cos(pi/2-a)=sin(a)
sin(pi/2+a)=cos(a)
cos(pi/2+a)=-sin(a)
sin(pi-a)=sin(a)
cos(pi-a)=-cos(a)
sin(pi+a)=-sin(a)
cos(pi+a)=-cos(a)
tga=tana=sina/cosa
万能公式
sin(a)= (2tan(a/2))/(1+tan^2(a/2))
cos(a)= (1-tan^2(a/2))/(1+tan^2(a/2))
tan(a)= (2tan(a/2))/(1-tan^2(a/2))
其它公式
a*sin(a)+b*cos(a)=sqrt(a^2+b^2)sin(a+c) [其中，tan(c)=b/a]
a*sin(a)-b*cos(a)=sqrt(a^2+b^2)cos(a-c) [其中，tan(c)=a/b]
1+sin(a)=(sin(a/2)+cos(a/2))^2
1-sin(a)=(sin(a/2)-cos(a/2))^2
其他非重点三角函数
csc(a)=1/sin(a)
sec(a)=1/cos(a)
双曲函数
sinh(a)=(e^a-e^(-a))/2
cosh(a)=(e^a+e^(-a))/2
tgh(a)=sinh(a)/cosh(a)) 望采纳
sin2a=2sina*cosa cos2a=cosa^2-sina^2=1-2sina^2
cosacosb=(cos(a+b)+cos（a-b))/2
绛旓細鎺ㄥ鍏紡锛(a+b+c)/(sinA+sinB+sinC)=2R(鍏朵腑锛孯涓哄鎺ュ渾鍗婂緞)鐢辨寮﹀畾鐞嗘湁 a/sinA=b/sinB=c/sinC=2R 鎵浠 a=2R*sinA b=2R*sinB c=2R*sinC 鍔犺捣鏉+b+c=2R*(sinA+sinB+sinC)甯﹀叆 (a+b+c)/(sinA+sinB+sinC)=2R*(sinA+sinB+sinC)/(sinA+sinB+sinC)=2R 涓よ鍜屽叕寮 sin...
绛旓細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)鍊嶈鍏紡 tan2a=2tana/[1-(tana)^2]cos2a=(cosa)^2-(...
绛旓細cosαsinβ = [sin(α+β)-sin(α-β)]/2 鍏紡涓锛氳α涓轰换鎰忚,缁堣竟鐩稿悓鐨勮鐨勫悓涓涓夎鍑芥暟鐨鍊肩浉绛夛細sin(2kπ+α)= sinαcos(2kπ+α)= cosαtan(2kπ+α)= tanαcot(2kπ+α)= cotα...
绛旓細缁堣竟鐩稿悓鐨勮鐨勫悓涓涓夎鍑芥暟鐨鍊肩浉绛夛細sin锛2k蟺锛嬑憋級锛漵in伪cos锛2k蟺锛嬑憋級锛漜os伪tan锛2k蟺锛嬑憋級锛漷an伪cot锛2k蟺锛嬑憋級锛漜ot伪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锛埾锛嬑憋級锛濓紞sin伪cos锛埾锛嬑憋級锛濓紞cos伪tan锛埾锛嬑憋級锛漷an伪cot锛...
绛旓細楂樹腑鏁板蹇呬慨4涓夎鍑芥暟鍏紡澶у叏 5 4涓洖绛 #鐑# 鍏徃閭ｄ簺璁炬柦鍙互鎻愰珮鍛樺伐骞哥鎰? 123鍟015 2012-02-18 路 TA鑾峰緱瓒呰繃166涓禐 鐭ラ亾绛斾富 鍥炵瓟閲:63 閲囩撼鐜:0% 甯姪鐨勪汉:33涓 鎴戜篃鍘荤瓟棰樿闂釜浜洪〉 鍏虫敞 灞曞紑鍏ㄩ儴 鍏紡涓: 璁疚变负浠绘剰瑙,缁堣竟鐩稿悓鐨勮鐨勫悓涓涓夎鍑芥暟鐨勫肩浉绛: sin(2k蟺+...
绛旓細-bc =(b+c)²-3bc =16-3bc鈮16-12=4 鍗冲綋b+c=4鏃讹紝a²鈮4锛屾墍浠鐨勬渶灏忓艰繕鏄2锛3锛夊綋b²+c²=4鏃讹紝鍥犱负b²+c²鈮2bc 鎵浠c鈮2锛-bc鈮-2 鎵浠²=b²+c²-bc =4-bc 鈮4-2=2 鍗砤²鈮2 鎵浠鈮モ垰2 ...
绛旓細路鍊嶈鍏紡锛歴in(2伪)=2sin伪路cos伪=2/(tan伪+cot伪)cos(2伪)=cos^2(伪)-sin^2(伪)=2cos^2(伪)-1=1-2sin^2(伪)tan(2伪)=2tan伪/[1-tan^2(伪)]路涓夊嶈鍏紡锛歴in3伪=3sin伪-4sin^3(伪)cos3伪=4cos^3(伪)-3cos伪 路鍗婅鍏紡锛歴in(伪/2)=姝ｈ礋鈭((1-cos伪)/2...
绛旓細鍏紡涓夛細 浠绘剰瑙捨变笌 -伪鐨涓夎鍑芥暟鍊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 鍏紡鍥锛 鍒╃敤鍏紡浜屽拰鍏紡涓夊彲浠ュ緱鍒跋-伪涓幬辩殑涓夎鍑芥暟鍊间箣闂寸殑鍏崇郴锛歴in锛埾锛嵨憋級=sin伪 cos锛埾锛嵨憋級=锛峜os伪 tan锛埾锛嵨憋級=锛峵an伪 cot锛...
绛旓細cosx/3鍙栧緱鏈灏忓-1锛寉鍙栧緱鏈澶у3 鈭磞鍙栧緱鏈灏忓3锛屾鏃秞闆嗗悎锝泋|x=6k蟺+2蟺,k鈭圸} 鈶 鐢2k蟺+蟺/2鈮2X+4鍒嗕箣蟺鈮2k蟺+3蟺/2 寰2k蟺+蟺/4鈮2X鈮2k蟺+5蟺/4锛宬鈭圸 鈭磌蟺+蟺/8鈮2X鈮蟺+5蟺/8锛宬鈭圸 鈭鍑芥暟鍗曡皟閫掑噺鍖洪棿 [k蟺+蟺/8,k蟺+5蟺/8]锛宬鈭圸 ...
绛旓細1,3涓哄拰瑙鍏紡鐨閫嗚瘉鏄庯紝鍙互鍊熷姪浜庡钩闈㈠悜閲忔暟閲忕Н璇佹槑锛屼篃鍙互鍒╃敤涓夎鍑芥暟鐨勫畾涔夎瘉鏄庛備綘鍏堢湅鐪嬪拰瑙掑叕寮忥紝楂樹腑鏁板蹇呬慨4绗3绔 2锛屼綘濂藉儚鍐欓敊浜嗐傚簲璇ユ槸1+sin2A鐨勪簩娆℃柟鏍=|sinA+cosA|锛岃繖閲屾槸1=(sinA)骞虫柟锛嬶紙cosA)骞虫柟,鍐2sinA*cosA=sin2A锛屽氨鏄畬鍏ㄥ钩鏂瑰拰銆
