高考数学
\u9ad8\u8003\u6570\u5b66\u6211\u4eca\u5e74\u5927\u56db\uff0c\u5728\u9ad8\u8003\u6570\u5b66\u65b9\u9762\u8fd8\u6709\u70b9\u7ecf\u9a8c\u3002\u73b0\u5728\u8ba9\u4f60\u518d\u4e00\u70b9\u70b9\u770b\u4e66\u672c\uff0c\u4e00\u662f\u65f6\u95f4\u4e0d\u5141\u8bb8\u4e86\uff0c\u4e8c\u662f\u4f60\u53ef\u80fd\u4f1a\u5fc3\u614c\uff0c\u8fdb\u800c\u5f71\u54cd\u4fe1\u5fc3\u3002
\u6700\u597d\u7684\u65b9\u6cd5\u5c31\u662f\u505a\u5386\u5e74\u771f\u9898\uff0c\u5341\u5e74\u7684\u771f\u9898\u51e0\u4e4e\u6db5\u76d6\u4e86\u6240\u6709\u7684\u8003\u70b9\uff0c\u5728\u505a\u9898\u8fc7\u7a0b\u4e2d\u9047\u5230\u4e0d\u4f1a\u7684\u77e5\u8bc6\u70b9\uff0c\u8d76\u7d27\u641e\u61c2\uff0c\u81ea\u5df1\u641e\u4e0d\u660e\u767d\uff0c\u8d76\u5feb\u95ee\u8001\u5e08\u6216\u540c\u5b66\uff0c\u4e0d\u8981\u81ea\u5df1\u6b7b\u6263\uff0c\u73b0\u5728\u65f6\u95f4\u6700\u5b9d\u8d35\u3002\u5341\u5e74\u771f\u9898\u505a\u4e00\u904d\u662f\u4e0d\u591f\uff0c\u81f3\u5c11\u4e24\u904d\u3002\u6709\u7b2c\u4e00\u904d\u7684\u57fa\u7840\uff0c\u7b2c\u4e8c\u904d\u770b\u7684\u65f6\u5019\u80af\u5b9a\u5f88\u5feb\u4e86\u300270\u5929\u4e24\u904d\u5341\u5e74\u771f\u9898\uff0c\u65f6\u95f4\u8fd8\u662f\u5f88\u5145\u88d5\u7684\u3002\u76f8\u4fe1\u81ea\u5df1\u3002
\u8fd9\u662f\u6211\u7684\u9ad8\u8003\u7ecf\u9a8c\uff0c\u5e0c\u671b\u5bf9\u4f60\u6709\u5e2e\u52a9\u3002
\u4e00\u3001\u9ad8\u4e2d\u6570\u5b66\u8bf1\u5bfc\u516c\u5f0f\u5168\u96c6\uff1a
\u5e38\u7528\u7684\u8bf1\u5bfc\u516c\u5f0f\u6709\u4ee5\u4e0b\u51e0\u7ec4\uff1a
\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 \uff08k\u2208Z\uff09
cos\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcos\u03b1 \uff08k\u2208Z\uff09
tan\uff082k\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1 \uff08k\u2208Z\uff09
cot\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcot\u03b1 \uff08k\u2208Z\uff09
\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)
\u6ce8\u610f\uff1a\u5728\u505a\u9898\u65f6\uff0c\u5c06a\u770b\u6210\u9510\u89d2\u6765\u505a\u4f1a\u6bd4\u8f83\u597d\u505a\u3002
\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\u4e8e\u03c0/2*k \u00b1\u03b1(k\u2208Z)\u7684\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
\uff03
\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(\u4f59\u5272)\uff1b\u4e09\u4e24\u5207\uff1b\u56db\u4f59\u5f26(\u6b63\u5272)\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
\u4e0a\u8ff0\u8bb0\u5fc6\u53e3\u8bc0,\u4e00\u5168\u6b63,\u4e8c\u6b63\u5f26,\u4e09\u5185\u5207,\u56db\u4f59\u5f26
\uff03
\u8fd8\u6709\u4e00\u79cd\u6309\u7167\u51fd\u6570\u7c7b\u578b\u5206\u8c61\u9650\u5b9a\u6b63\u8d1f\uff1a
\u51fd\u6570\u7c7b\u578b \u7b2c\u4e00\u8c61\u9650 \u7b2c\u4e8c\u8c61\u9650 \u7b2c\u4e09\u8c61\u9650 \u7b2c\u56db\u8c61\u9650
\u6b63\u5f26 ...........\uff0b............\uff0b............\u2014............\u2014........
\u4f59\u5f26 ...........\uff0b............\u2014............\u2014............\uff0b........
\u6b63\u5207 ...........\uff0b............\u2014............\uff0b............\u2014........
\u4f59\u5207 ...........\uff0b............\u2014............\uff0b............\u2014........
\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u57fa\u672c\u5173\u7cfb
\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
\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\uff08\u03b1\uff0b\u03b2\uff09\uff1d(tan\u03b1+tan\u03b2)\uff0f(1-tan\u03b1tan\u03b2)
tan\uff08\u03b1\uff0d\u03b2\uff09\uff1d(tan\u03b1\uff0dtan\u03b2)\uff0f(1\uff0btan\u03b1\u00b7tan\u03b2)
\u4e8c\u500d\u89d2\u516c\u5f0f
\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)
tan2\u03b1\uff1d2tan\u03b1/[1\uff0dtan^2(\u03b1)]
\u534a\u89d2\u516c\u5f0f
\u534a\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f\uff08\u964d\u5e42\u6269\u89d2\u516c\u5f0f\uff09
sin^2(\u03b1/2)\uff1d(1\uff0dcos\u03b1)\uff0f2
cos^2(\u03b1/2)\uff1d(1\uff0bcos\u03b1)\uff0f2
tan^2(\u03b1/2)\uff1d(1\uff0dcos\u03b1)\uff0f(1\uff0bcos\u03b1)
\u53e6\u4e5f\u6709tan(\u03b1/2)=(1\uff0dcos\u03b1)/sin\u03b1=sin\u03b1/(1+cos\u03b1)
\u4e07\u80fd\u516c\u5f0f
\u4e07\u80fd\u516c\u5f0f
sin\u03b1=2tan(\u03b1/2)/[1+tan^2(\u03b1/2)]
cos\u03b1=[1-tan^2(\u03b1/2)]/[1+tan^2(\u03b1/2)]
tan\u03b1=2tan(\u03b1/2)/[1-tan^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\uff1d2tan\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
\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
tan3\u03b1\uff1d[3tan\u03b1\uff0dtan^3(\u03b1)]\uff0f[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^3(\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
\u2605\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
\u2605\u53e6\u5916\u7684\u8bb0\u5fc6\u65b9\u6cd5:
\u6b63\u5f26\u4e09\u500d\u89d2: \u5c71\u65e0\u53f8\u4ee4 (\u8c10\u97f3\u4e3a \u4e09\u65e0\u56db\u7acb) \u4e09\u6307\u7684\u662f"3\u500d"sin\u03b1, \u65e0\u6307\u7684\u662f\u51cf\u53f7, \u56db\u6307\u7684\u662f"4\u500d", \u7acb\u6307\u7684\u662fsin\u03b1\u7acb\u65b9
\u4f59\u5f26\u4e09\u500d\u89d2: \u53f8\u4ee4\u65e0\u5c71 \u4e0e\u4e0a\u540c\u7406
\u548c\u5dee\u5316\u79ef\u516c\u5f0f
\u4e09\u89d2\u51fd\u6570\u7684\u548c\u5dee\u5316\u79ef\u516c\u5f0f
sin\u03b1\uff0bsin\u03b2\uff1d2sin[(\u03b1\uff0b\u03b2)/2]\u00b7cos[(\u03b1\uff0d\u03b2)/2]
sin\u03b1\uff0dsin\u03b2\uff1d2cos[(\u03b1\uff0b\u03b2)/2]\u00b7sin[(\u03b1\uff0d\u03b2)/2]
cos\u03b1\uff0bcos\u03b2\uff1d2cos[(\u03b1\uff0b\u03b2)/2]\u00b7cos[(\u03b1\uff0d\u03b2)/2]
cos\u03b1\uff0dcos\u03b2\uff1d\uff0d2sin[(\u03b1\uff0b\u03b2)/2]\u00b7sin[(\u03b1\uff0d\u03b2)/2]
\u79ef\u5316\u548c\u5dee\u516c\u5f0f
\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5316\u548c\u5dee\u516c\u5f0f
sin\u03b1 \u00b7cos\u03b2\uff1d0.5[sin(\u03b1\uff0b\u03b2)\uff0bsin(\u03b1\uff0d\u03b2)]
cos\u03b1 \u00b7sin\u03b2\uff1d0.5[sin(\u03b1\uff0b\u03b2)\uff0dsin(\u03b1\uff0d\u03b2)]
cos\u03b1 \u00b7cos\u03b2\uff1d0.5[cos(\u03b1\uff0b\u03b2)\uff0bcos(\u03b1\uff0d\u03b2)]
sin\u03b1 \u00b7sin\u03b2\uff1d\uff0d0.5[cos(\u03b1\uff0b\u03b2)\uff0dcos(\u03b1\uff0d\u03b2)]
\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)
线与线是0~90
因为直线没有方向,所以总是取小角,0到90
而向量比如力,有方向,方向不同,合向量不同,所以是0到180
线面角0~90 同线线角,取小角
二面角0~180
因为二个面都不是整个平面,所以是0~180
向量是0~180
线与线是0~90
而向量,有方向,方向不同,合向量不同,所以是0到180
线面角0~90 同线线角,取小角
二面角0~180
因为二个面都不是整个平面,所以是0~180
向量：0,180 左右包含
线:0,90
线面：0,90
二面:0,180
绛旓細楂樿冩暟瀛涓昏鑰冨療浠ヤ笅鍐呭锛氫竴銆佸熀纭鐭ヨ瘑 楂樿冩暟瀛﹀熀纭鐭ヨ瘑鐐瑰寘鎷疄鏁般佷唬鏁般佸嚑浣曘佹鐜囩粺璁＄瓑銆傚疄鏁伴儴鍒嗕富瑕佽冨療鏈夌悊鏁板拰鏃犵悊鏁扮殑杩愮畻锛涗唬鏁伴儴鍒嗘秹鍙婂嚱鏁般佹暟鍒椼佷笉绛夊紡绛夌煡璇嗙偣锛涘嚑浣曢儴鍒嗗垯鍖呮嫭骞抽潰瑙ｆ瀽鍑犱綍鍜岀珛浣撳嚑浣曪紱姒傜巼缁熻閮ㄥ垎涓昏鑰冨療鏁版嵁鐨勬敹闆嗐佹暣鐞嗗拰鍒嗘瀽銆備簩銆佽В棰樻妧鑳 楂樿冩暟瀛︽敞閲嶈冨療瀛︾敓鐨勮В棰...
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绛旓細楂樿冩暟瀛蹇呰冨叕寮忓涓嬶細1銆佹姏鐗╃嚎锛歽=ax*+bx+c灏辨槸y绛変簬ax鐨勫钩鏂瑰姞涓奲x鍐嶅姞涓奵銆俛>0鏃讹紝鎶涚墿绾垮紑鍙ｅ悜涓;a<0鏃舵姏鐗╃嚎寮鍙ｅ悜涓;c=0鏃舵姏鐗╃嚎缁忚繃鍘熺偣;b=0鏃舵姏鐗╃嚎瀵圭О杞翠负y杞淬2銆侀《鐐瑰紡y=a(x+h)*+k灏辨槸y绛変簬a涔樹互(x+h)鐨勫钩鏂+k锛-h鏄《鐐瑰潗鏍囩殑x锛宬鏄《鐐瑰潗鏍囩殑y锛屼竴鑸敤浜庢眰鏈澶у...
绛旓細楂樿蹇呭鏁板鍏紡锛1銆佷笁瑙掑嚱鏁帮細sin(a+b)=sin(a)cos(b)+cos(a)sin(b)銆乧os(a+b)=cos(a)cos(b)-sin(a)sin(b)銆乼an(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))銆乻in^2(a)+cos^2(a)=1銆1+tan^2(a)=sec^2(a)銆1+cot^2(a)=csc^2(a)2銆佸钩闈㈠嚑浣曪細鍕捐偂瀹氱悊锛歛^...
绛旓細楂樿冩暟瀛鑰冭瘯鏃堕棿涓120鍒嗛挓,涔熷氨鏄2灏忔椂銆3+鏂囩患/鐞嗙患浼犵粺楂樿冩椂闂村畨鎺掞細銆愯鏂囥6鏈7鏃9:00-11:30鑰冭瘯鏃堕暱锛150鍒嗛挓 銆愭暟瀛︺6鏈7鏃15:00-17:00鑰冭瘯鏃堕暱锛120鍒嗛挓 銆愭枃缁/鐞嗙患銆6鏈8鏃9:00-11:30鑰冭瘯鏃堕暱锛150鍒嗛挓 銆愬璇6鏈8鏃15:00-17:00鑰冭瘯鏃堕暱锛120鍒嗛挓 3+1+2鏂伴珮鑰冩椂闂村畨鎺...
