高中必修四数学公式有哪些 高中数学必修四所有公式
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a^2-b^2=(a+b)(a-b)
a^3+b^3=(a+b)(a^2-ab+b^2)
a^3-b^3=(a-b(a^2+ab+b^2)
\u4e09\u89d2\u4e0d\u7b49\u5f0f |a+b|\u2264|a|+|b| |a-b|\u2264|a|+|b| |a|\u2264b-b\u2264a\u2264b
|a-b|\u2265|a|-|b| -|a|\u2264a\u2264|a|
\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u7684\u89e3 -b+\u221a(b^2-4ac)/2a -b-\u221a(b^2-4ac)/2a
\u6839\u4e0e\u7cfb\u6570\u7684\u5173\u7cfb X1+X2=-b/a X1*X2=c/a \u6ce8\uff1a\u97e6\u8fbe\u5b9a\u7406
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b^2-4ac=0 \u6ce8\uff1a\u65b9\u7a0b\u6709\u4e24\u4e2a\u76f8\u7b49\u7684\u5b9e\u6839
b^2-4ac>0 \u6ce8\uff1a\u65b9\u7a0b\u6709\u4e24\u4e2a\u4e0d\u7b49\u7684\u5b9e\u6839
b^2-4ac<0 \u6ce8\uff1a\u65b9\u7a0b\u6ca1\u6709\u5b9e\u6839\uff0c\u6709\u5171\u8f6d\u590d\u6570\u6839
\u4e09\u89d2\u51fd\u6570\u516c\u5f0f
\u4e24\u89d2\u548c\u516c\u5f0f
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)
\u500d\u89d2\u516c\u5f0f
tan2A=2tanA/[1-(tanA)^2]
cos2a=(cosa)^2-(sina)^2=2(cosa)^2 -1=1-2(sina)^2
\u534a\u89d2\u516c\u5f0f
sin(A/2)=\u221a((1-cosA)/2) sin(A/2)=-\u221a((1-cosA)/2)
cos(A/2)=\u221a((1+cosA)/2) cos(A/2)=-\u221a((1+cosA)/2)
tan(A/2)=\u221a((1-cosA)/((1+cosA)) tan(A/2)=-\u221a((1-cosA)/((1+cosA))
cot(A/2)=\u221a((1+cosA)/((1-cosA)) cot(A/2)=-\u221a((1+cosA)/((1-cosA))
\u548c\u5dee\u5316\u79ef
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
\u67d0\u4e9b\u6570\u5217\u524dn\u9879\u548c
1+2+3+4+5+6+7+8+9+\u2026+n=n(n+1)/2
1+3+5+7+9+11+13+15+\u2026+(2n-1)=n2 \u001e
2+4+6+8+10+12+14+\u2026+(2n)=n(n+1) 5
1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+\u2026+n^2=n(n+1)(2n+1)/6
1^3+2^3+3^3+4^3+5^3+6^3+\u2026n^3=n2(n+1)2/4
1*2+2*3+3*4+4*5+5*6+6*7+\u2026+n(n+1)=n(n+1)(n+2)/3
\u6b63\u5f26\u5b9a\u7406 a/sinA=b/sinB=c/sinC=2R \u6ce8\uff1a \u5176\u4e2d R \u8868\u793a\u4e09\u89d2\u5f62\u7684\u5916\u63a5\u5706\u534a\u5f84
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\u5706\u7684\u6807\u51c6\u65b9\u7a0b (x-a)^2+(y-b)^2=^r2 \u6ce8\uff1a\uff08a,b\uff09\u662f\u5706\u5fc3\u5750\u6807 \u001f
\u5706\u7684\u4e00\u822c\u65b9\u7a0b x^2+y^2+Dx+Ey+F=0 \u6ce8\uff1aD^2+E^2-4F>0
\u629b\u7269\u7ebf\u6807\u51c6\u65b9\u7a0b y^2=2px y^2=-2px x^2=2py x^2=-2py
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\u5f27\u957f\u516c\u5f0f l=a*r a\u662f\u5706\u5fc3\u89d2\u7684\u5f27\u5ea6\u6570r >0 \u6247\u5f62\u9762\u79ef\u516c\u5f0f s=1/2*l*r
\u9525\u4f53\u4f53\u79ef\u516c\u5f0f V=1/3*S*H \u5706\u9525\u4f53\u4f53\u79ef\u516c\u5f0f V=1/3*pi*r2h
\u659c\u68f1\u67f1\u4f53\u79ef V=S'L \u6ce8\uff1a\u5176\u4e2d,S'\u662f\u76f4\u622a\u9762\u9762\u79ef\uff0c L\u662f\u4fa7\u68f1\u957f
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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
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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
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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
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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
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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)
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\u5bf9\u4e8ek\u00b7\u03c0/2\u00b1\u03b1(k\u2208Z)\u7684\u4e2a\u4e09\u89d2\u51fd\u6570\u503c\uff0c
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\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
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\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
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\u5176\u4ed6\u4e09\u89d2\u51fd\u6570\u77e5\u8bc6\uff1a
\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u57fa\u672c\u5173\u7cfb
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\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
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\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
\u5411\u91cf\u7684\u52a0\u6cd5\u8fd0\u7b97\u3001\u51cf\u6cd5\u8fd0\u7b97\u3001\u6570\u4e58\u8fd0\u7b97\u7edf\u79f0\u7ebf\u6027\u8fd0\u7b97\u3002
\u5411\u91cf\u7684\u6570\u91cf\u79ef
\u5df2\u77e5\u4e24\u4e2a\u975e\u96f6\u5411\u91cfa\u3001b\uff0c\u90a3\u4e48|a||b|cos \u03b8\u53eb\u505aa\u4e0eb\u7684\u6570\u91cf\u79ef\u6216\u5185\u79ef\uff0c\u8bb0\u4f5ca•b\uff0c\u03b8\u662fa\u4e0eb\u7684\u5939\u89d2\uff0c|a|cos \u03b8\uff08|b|cos \u03b8\uff09\u53eb\u505a\u5411\u91cfa\u5728b\u65b9\u5411\u4e0a\uff08b\u5728a\u65b9\u5411\u4e0a\uff09\u7684\u6295\u5f71\u3002\u96f6\u5411\u91cf\u4e0e\u4efb\u610f\u5411\u91cf\u7684\u6570\u91cf\u79ef\u4e3a0\u3002
a•b\u7684\u51e0\u4f55\u610f\u4e49\uff1a\u6570\u91cf\u79efa•b\u7b49\u4e8ea\u7684\u957f\u5ea6|a|\u4e0eb\u5728a\u7684\u65b9\u5411\u4e0a\u7684\u6295\u5f71|b|cos \u03b8\u7684\u4e58\u79ef\u3002
\u4e24\u4e2a\u5411\u91cf\u7684\u6570\u91cf\u79ef\u7b49\u4e8e\u5b83\u4eec\u5bf9\u5e94\u5750\u6807\u7684\u4e58\u79ef\u7684\u548c\u3002
\u516c\u5f0f\u4e00:
\u8bbe\u03b1\u4e3a\u4efb\u610f\u89d2\uff0c\u7ec8\u8fb9\u76f8\u540c\u7684\u89d2\u7684\u540c\u4e00\u4e09\u89d2\u51fd\u6570\u7684\u503c\u76f8\u7b49:
sin(2k\u03c0+\u03b1)=sin\u03b1
cos(2k\u03c0+\u03b1)=cos\u03b1
tan(2k\u03c0+\u03b1)=tan\u03b1
cot(2k\u03c0+\u03b1)=cot\u03b1
\u516c\u5f0f\u4e8c:
\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:
sin(\u03c0+\u03b1)=-sin\u03b1
cos(\u03c0+\u03b1)=-cos\u03b1
tan(\u03c0+\u03b1)=tan\u03b1
cot(\u03c0+\u03b1)=cot\u03b1
\u516c\u5f0f\u4e09:
\u4efb\u610f\u89d2\u03b1\u4e0e
-\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb:
sin(-\u03b1)=-sin\u03b1
cos(-\u03b1)=cos\u03b1
tan(-\u03b1)=-tan\u03b1
cot(-\u03b1)=-cot\u03b1
\u516c\u5f0f\u56db:
\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:
sin(\u03c0-\u03b1)=sin\u03b1
cos(\u03c0-\u03b1)=-cos\u03b1
tan(\u03c0-\u03b1)=-tan\u03b1
cot(\u03c0-\u03b1)=-cot\u03b1
\u516c\u5f0f\u4e94:
\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:
sin(2\u03c0-\u03b1)=-sin\u03b1
cos(2\u03c0-\u03b1)=cos\u03b1
tan(2\u03c0-\u03b1)=-tan\u03b1
cot(2\u03c0-\u03b1)=-cot\u03b1
\u516c\u5f0f\u516d:
\u03c0/2\u00b1\u03b1\u53ca3\u03c0/2\u00b1\u03b1\u4e0e\u03b1\u7684\u4e09\u89d2\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u5173\u7cfb:
sin(\u03c0/2+\u03b1)=cos\u03b1
cos(\u03c0/2+\u03b1)=-sin\u03b1
tan(\u03c0/2+\u03b1)=-cot\u03b1
cot(\u03c0/2+\u03b1)=-tan\u03b1
sin(\u03c0/2-\u03b1)=cos\u03b1
cos(\u03c0/2-\u03b1)=sin\u03b1
tan(\u03c0/2-\u03b1)=cot\u03b1
cot(\u03c0/2-\u03b1)=tan\u03b1
sin(3\u03c0/2+\u03b1)=-cos\u03b1
cos(3\u03c0/2+\u03b1)=sin\u03b1
tan(3\u03c0/2+\u03b1)=-cot\u03b1
cot(3\u03c0/2+\u03b1)=-tan\u03b1
sin(3\u03c0/2-\u03b1)=-cos\u03b1
cos(3\u03c0/2-\u03b1)=-sin\u03b1
tan(3\u03c0/2-\u03b1)=cot\u03b1
cot(3\u03c0/2-\u03b1)=tan\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:
\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;
\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.
(\u5947\u53d8\u5076\u4e0d\u53d8)
\u7136\u540e\u5728\u524d\u9762\u52a0\u4e0a\u628a\u03b1\u770b\u6210\u9510\u89d2\u65f6\u539f\u51fd\u6570\u503c\u7684\u7b26\u53f7\u3002
(\u7b26\u53f7\u770b\u8c61\u9650)
\u4f8b\u5982:
sin(2\u03c0-\u03b1)=sin(4\u00b7\u03c0/2-\u03b1)\uff0ck=4\u4e3a\u5076\u6570\uff0c\u6240\u4ee5\u53d6sin\u03b1\u3002
\u5f53\u03b1\u662f\u9510\u89d2\u65f6\uff0c2\u03c0-\u03b1\u2208(270\u00b0\uff0c360\u00b0)\uff0csin(2\u03c0-\u03b1)\uff1c0\uff0c\u7b26\u53f7\u4e3a\u201c-\u201d\u3002
\u6240\u4ee5sin(2\u03c0-\u03b1)=-sin\u03b1
\u4e0a\u8ff0\u7684\u8bb0\u5fc6\u53e3\u8bc0\u662f:
\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(k\u2208Z)\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;\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;\u4e8c\u6b63\u5f26;\u4e09\u4e3a\u5207;\u56db\u4f59\u5f26\u201d.
\u8fd9\u5341\u4e8c\u5b57\u53e3\u8bc0\u7684\u610f\u601d\u5c31\u662f\u8bf4:
\u7b2c\u4e00\u8c61\u9650\u5185\u4efb\u4f55\u4e00\u4e2a\u89d2\u7684\u56db\u79cd\u4e09\u89d2\u51fd\u6570\u503c\u90fd\u662f\u201c+\u201d;
\u7b2c\u4e8c\u8c61\u9650\u5185\u53ea\u6709\u6b63\u5f26\u662f\u201c+\u201d\uff0c\u5176\u4f59\u5168\u90e8\u662f\u201c-\u201d;
\u7b2c\u4e09\u8c61\u9650\u5185\u5207\u51fd\u6570\u662f\u201c+\u201d\uff0c\u5f26\u51fd\u6570\u662f\u201c-\u201d;
\u7b2c\u56db\u8c61\u9650\u5185\u53ea\u6709\u4f59\u5f26\u662f\u201c+\u201d\uff0c\u5176\u4f59\u5168\u90e8\u662f\u201c-\u201d.
\u5176\u4ed6\u4e09\u89d2\u51fd\u6570\u77e5\u8bc6:
\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=1
sin\u03b1
\u00b7csc\u03b1=1
cos\u03b1
\u00b7sec\u03b1=1
\u5546\u7684\u5173\u7cfb:
sin\u03b1/cos\u03b1=tan\u03b1=sec\u03b1/csc\u03b1
cos\u03b1/sin\u03b1=cot\u03b1=csc\u03b1/sec\u03b1
\u5e73\u65b9\u5173\u7cfb:
sin^2(\u03b1)+cos^2(\u03b1)=1
1+tan^2(\u03b1)=sec^2(\u03b1)
1+cot^2(\u03b1)=csc^2(\u03b1)
\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u5173\u7cfb\u516d\u89d2\u5f62\u8bb0\u5fc6\u6cd5
\u516d\u89d2\u5f62\u8bb0\u5fc6\u6cd5:(\u53c2\u770b\u56fe\u7247\u6216\u53c2\u8003\u8d44\u6599\u94fe\u63a5)
\u6784\u9020\u4ee5"\u4e0a\u5f26\u3001\u4e2d\u5207\u3001\u4e0b\u5272;\u5de6\u6b63\u3001\u53f3\u4f59\u3001\u4e2d\u95f41"\u7684\u6b63\u516d\u8fb9\u5f62\u4e3a\u6a21\u578b\u3002
(1)\u5012\u6570\u5173\u7cfb:\u5bf9\u89d2\u7ebf\u4e0a\u4e24\u4e2a\u51fd\u6570\u4e92\u4e3a\u5012\u6570;
(2)\u5546\u6570\u5173\u7cfb:\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
(\u4e3b\u8981\u662f\u4e24\u6761\u865a\u7ebf\u4e24\u7aef\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7684\u4e58\u79ef)\u3002\u7531\u6b64\uff0c\u53ef\u5f97\u5546\u6570\u5173\u7cfb\u5f0f\u3002
(3)\u5e73\u65b9\u5173\u7cfb:\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(\u03b1+\u03b2)=sin\u03b1cos\u03b2+cos\u03b1sin\u03b2
sin(\u03b1-\u03b2)=sin\u03b1cos\u03b2-cos\u03b1sin\u03b2
cos(\u03b1+\u03b2)=cos\u03b1cos\u03b2-sin\u03b1sin\u03b2
cos(\u03b1-\u03b2)=cos\u03b1cos\u03b2+sin\u03b1sin\u03b2
tan\u03b1+tan\u03b2
tan(\u03b1+\u03b2)=------
1-tan\u03b1
\u00b7tan\u03b2
tan\u03b1-tan\u03b2
tan(\u03b1-\u03b2)=------
1+tan\u03b1
\u00b7tan\u03b2
\u500d\u89d2\u516c\u5f0f
\u248a\u4e8c\u500d\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f(\u5347\u5e42\u7f29\u89d2\u516c\u5f0f)
sin2\u03b1=2sin\u03b1cos\u03b1
cos2\u03b1=cos^2(\u03b1)-sin^2(\u03b1)=2cos^2(\u03b1)-1=1-2sin^2(\u03b1)
2tan\u03b1
tan2\u03b1=-----
1-tan^2(\u03b1)
\u534a\u89d2\u516c\u5f0f
\u248b\u534a\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f(\u964d\u5e42\u6269\u89d2\u516c\u5f0f)
1-cos\u03b1
sin^2(\u03b1/2)=-----
2
1+cos\u03b1
cos^2(\u03b1/2)=-----
2
1-cos\u03b1
tan^2(\u03b1/2)=-----
1+cos\u03b1
\u4e07\u80fd\u516c\u5f0f
\u248c\u4e07\u80fd\u516c\u5f0f
2tan(\u03b1/2)
sin\u03b1=------
1+tan^2(\u03b1/2)
1-tan^2(\u03b1/2)
cos\u03b1=------
1+tan^2(\u03b1/2)
2tan(\u03b1/2)
tan\u03b1=------
1-tan^2(\u03b1/2)
\u4e07\u80fd\u516c\u5f0f\u63a8\u5bfc
\u9644\u63a8\u5bfc:
sin2\u03b1=2sin\u03b1cos\u03b1=2sin\u03b1cos\u03b1/(cos^2(\u03b1)+sin^2(\u03b1))......*\uff0c
(\u56e0\u4e3acos^2(\u03b1)+sin^2(\u03b1)=1)
\u518d\u628a*\u5206\u5f0f\u4e0a\u4e0b\u540c\u9664cos^2(\u03b1)\uff0c\u53ef\u5f97sin2\u03b1=tan2\u03b1/(1+tan^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=3sin\u03b1-4sin^3(\u03b1)
cos3\u03b1=4cos^3(\u03b1)-3cos\u03b1
3tan\u03b1-tan^3(\u03b1)
tan3\u03b1=------
1-3tan^2(\u03b1)
\u4e09\u500d\u89d2\u516c\u5f0f\u63a8\u5bfc
\u9644\u63a8\u5bfc:
tan3\u03b1=sin3\u03b1/cos3\u03b1
=(sin2\u03b1cos\u03b1+cos2\u03b1sin\u03b1)/(cos2\u03b1cos\u03b1-sin2\u03b1sin\u03b1)
=(2sin\u03b1cos^2(\u03b1)+cos^2(\u03b1)sin\u03b1-sin^3(\u03b1))/(cos^3(\u03b1)-cos\u03b1sin^2(\u03b1)-2sin^2(\u03b1)cos\u03b1)
\u4e0a\u4e0b\u540c\u9664\u4ee5cos^3(\u03b1)\uff0c\u5f97:
tan3\u03b1=(3tan\u03b1-tan^3(\u03b1))/(1-3tan^2(\u03b1))
sin3\u03b1=sin(2\u03b1+\u03b1)=sin2\u03b1cos\u03b1+cos2\u03b1sin\u03b1
=2sin\u03b1cos^2(\u03b1)+(1-2sin^2(\u03b1))sin\u03b1
=2sin\u03b1-2sin^3(\u03b1)+sin\u03b1-2sin^2(\u03b1)
=3sin\u03b1-4sin^3(\u03b1)
cos3\u03b1=cos(2\u03b1+\u03b1)=cos2\u03b1cos\u03b1-sin2\u03b1sin\u03b1
=(2cos^2(\u03b1)-1)cos\u03b1-2cos\u03b1sin^2(\u03b1)
=2cos^3(\u03b1)-cos\u03b1+(2cos\u03b1-2cos^3(\u03b1))
=4cos^3(\u03b1)-3cos\u03b1
\u5373
sin3\u03b1=3sin\u03b1-4sin^3(\u03b1)
cos3\u03b1=4cos^3(\u03b1)-3cos\u03b1
\u4e09\u500d\u89d2\u516c\u5f0f\u8054\u60f3\u8bb0\u5fc6
\u8bb0\u5fc6\u65b9\u6cd5:\u8c10\u97f3\u3001\u8054\u60f3
\u6b63\u5f26\u4e09\u500d\u89d2:3\u5143
\u51cf
4\u51433\u89d2(\u6b20\u503a\u4e86(\u88ab\u51cf\u6210\u8d1f\u6570)\uff0c\u6240\u4ee5\u8981\u201c\u6323\u94b1\u201d(\u97f3\u4f3c\u201c\u6b63\u5f26\u201d))
\u4f59\u5f26\u4e09\u500d\u89d2:4\u51433\u89d2
\u51cf
3\u5143(\u51cf\u5b8c\u4e4b\u540e\u8fd8\u6709\u201c\u4f59\u201d)
\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+\u03b2
\u03b1-\u03b2
sin\u03b1+sin\u03b2=2sin-----\u00b7cos----
2
2
\u03b1+\u03b2
\u03b1-\u03b2
sin\u03b1-sin\u03b2=2cos-----\u00b7sin-----
2
2
\u03b1+\u03b2
\u03b1-\u03b2
cos\u03b1+cos\u03b2=2cos------\u00b7cos------
2
2
\u03b1+\u03b2
\u03b1-\u03b2
cos\u03b1-cos\u03b2=-2sin------\u00b7sin------
2
2
\u79ef\u5316\u548c\u5dee\u516c\u5f0f
\u248f\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5316\u548c\u5dee\u516c\u5f0f
sin\u03b1
\u00b7cos\u03b2=0.5[sin(\u03b1+\u03b2)+sin(\u03b1-\u03b2)]
cos\u03b1
\u00b7sin\u03b2=0.5[sin(\u03b1+\u03b2)-sin(\u03b1-\u03b2)]
cos\u03b1
\u00b7cos\u03b2=0.5[cos(\u03b1+\u03b2)+cos(\u03b1-\u03b2)]
sin\u03b1
\u00b7sin\u03b2=-
0.5[cos(\u03b1+\u03b2)-cos(\u03b1-\u03b2)]
\u548c\u5dee\u5316\u79ef\u516c\u5f0f\u63a8\u5bfc
\u9644\u63a8\u5bfc:
\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+BC=AC\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\u6709:0+a=a+0=a\u3002
|a+b|\u2264|a|+|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-(-a)=a\uff0c\u96f6\u5411\u91cf\u7684\u76f8\u53cd\u5411\u91cf\u4ecd\u7136\u662f\u96f6\u5411\u91cf\u3002
(1)a+(-a)=(-a)+a=0(2)a-b=a+(-b)\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|=|\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:(1)(\u03bb\u03bc)a
=
\u03bb(\u03bca)(2)(\u03bb
+
\u03bc)a
=
\u03bba
+
\u03bca(3)\u03bb(a
\u00b1
b)
=
\u03bba
\u00b1
\u03bbb(4)(-\u03bb)a
=-(\u03bba)
=
\u03bb(-a)\u3002
\u5411\u91cf\u7684\u52a0\u6cd5\u8fd0\u7b97\u3001\u51cf\u6cd5\u8fd0\u7b97\u3001\u6570\u4e58\u8fd0\u7b97\u7edf\u79f0\u7ebf\u6027\u8fd0\u7b97\u3002
\u5411\u91cf\u7684\u6570\u91cf\u79ef
\u5df2\u77e5\u4e24\u4e2a\u975e\u96f6\u5411\u91cfa\u3001b\uff0c\u90a3\u4e48|a||b|cos
\u03b8\u53eb\u505aa\u4e0eb\u7684\u6570\u91cf\u79ef\u6216\u5185\u79ef\uff0c\u8bb0\u4f5ca•b\uff0c\u03b8\u662fa\u4e0eb\u7684\u5939\u89d2\uff0c|a|cos
\u03b8(|b|cos
\u03b8)\u53eb\u505a\u5411\u91cfa\u5728b\u65b9\u5411\u4e0a(b\u5728a\u65b9\u5411\u4e0a)\u7684\u6295\u5f71\u3002\u96f6\u5411\u91cf\u4e0e\u4efb\u610f\u5411\u91cf\u7684\u6570\u91cf\u79ef\u4e3a0\u3002
a•b\u7684\u51e0\u4f55\u610f\u4e49:\u6570\u91cf\u79efa•b\u7b49\u4e8ea\u7684\u957f\u5ea6|a|\u4e0eb\u5728a\u7684\u65b9\u5411\u4e0a\u7684\u6295\u5f71|b|cos
\u03b8\u7684\u4e58\u79ef\u3002
\u4e24\u4e2a\u5411\u91cf\u7684\u6570\u91cf\u79ef\u7b49\u4e8e\u5b83\u4eec\u5bf9\u5e94\u5750\u6807\u7684\u4e58\u79ef\u7684\u548c\u3002
一、两角和差公式
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)
二、用以上公式可推出下列二倍角公式
tan2A=2tanA/[1-(tanA)^2]
cos2a=(cosa)^2-(sina)^2=2(cosa)^2 -1=1-2(sina)^2
sin2A=2sinA*cosA
三、半角的只需记住这个：
tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA)
四、用二倍角中的余弦可推出降幂公式
(sinA)^2=(1-cos2A)/2
(cosA)^2=(1+cos2A)/2
五、用以上降幂公式可推出以下常用的化简公式
1-cosA=sin^(A/2)*2
1-sinA=cos^(A/2)*2
绛旓細楂樹腑蹇呬慨鍥涙暟瀛﹀叕寮忎富瑕佸寘鎷笁瑙掑嚱鏁板叕寮忋佸悜閲忓叕寮忎互鍙婂鏁板叕寮忕瓑銆備笁瑙掑嚱鏁板叕寮忔槸楂樹腑蹇呬慨鍥涙暟瀛︿腑鐨勯噸鐐逛箣涓銆傛寮﹀嚱鏁般佷綑寮﹀嚱鏁板拰姝ｅ垏鍑芥暟鏄笁瑙掑嚱鏁扮殑鍩烘湰鍐呭銆傚叾涓紝姝ｅ鸡鍑芥暟sin(x)琛ㄧず涓涓x鐨勬寮﹀硷紝浣欏鸡鍑芥暟cos(x)琛ㄧず涓涓x鐨勪綑寮﹀硷紝姝ｅ垏鍑芥暟tan(x)琛ㄧず涓涓x鐨勬鍒囧笺傛澶栵紝杩樻湁鍜屽樊...
绛旓細涓銆佷袱瑙掑拰宸叕寮 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)浜屻佺敤浠ヤ笂鍏紡鍙帹鍑轰笅鍒椾簩鍊嶈鍏紡 tan2A=2tanA/[1-(tanA)...
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绛旓細姝ｅ鸡鍏紡锛歴in = sin伪cos尾 + cos伪sin尾锛泂in = sin伪cos尾 - cos伪sin尾銆備綑寮﹀叕寮忥細cos = cos伪cos尾 - sin伪sin尾锛沜os = cos伪cos尾 + sin伪sin尾銆2. 鍊嶈鍏紡 姝ｅ鸡鍏紡锛歴in2伪 = 2sin伪cos伪锛涗綑寮﹀叕寮忥細cos2伪 = cos²伪 - sin²伪銆傛鍒囧叕寮忥細tan2伪 = 2...
绛旓細鍏紡涓锛氳伪涓轰换鎰忚锛岀粓杈圭浉鍚岀殑瑙掔殑鍚屼竴涓夎鍑芥暟鐨勫肩浉绛夛細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...
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绛旓細OP=(OP1+位OP2)(1+位)锛涳紙瀹氭瘮鍒嗙偣鍚戦噺鍏紡锛墄=(x1+位x2)/(1+位),y=(y1+位y2)/(1+位).锛堝畾姣斿垎鐐瑰潗鏍囧叕寮忥級鎴戜滑鎶婁笂闈㈢殑寮忓瓙鍙仛鏈夊悜绾挎P1P2鐨勫畾姣斿垎鐐瑰叕寮 涓夌偣鍏辩嚎瀹氱悊 鑻C=位OA +渭OB ,涓斘+渭=1 ,鍒橝銆丅銆丆涓夌偣鍏辩嚎 涓夎褰㈤噸蹇冨垽鏂紡 鍦ㄢ柍ABC涓,鑻A +GB +GC...
绛旓細銆鍏紡涓銆戣伪涓轰换鎰忚锛岀粓杈圭浉鍚岀殑瑙掔殑鍚屼竴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)銆愬叕寮忎簩銆戣伪涓轰换鎰忚锛屜+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)=...
绛旓細閿ヤ綋浣撶Н鍏紡 V=1/3*S*H 鍦嗛敟浣撲綋绉叕寮 V=1/3*pi*r2h 鏂滄１鏌变綋绉 V=S'L 娉:鍏朵腑,S'鏄洿鎴潰闈㈢Н, L鏄晶妫遍暱 鏌变綋浣撶Н鍏紡 V=s*h 鍦嗘煴浣 V=pi*r2h 瀹氱悊: 1 杩囦袱鐐规湁涓斿彧鏈変竴鏉＄洿绾 2 涓ょ偣涔嬮棿绾挎鏈鐭 3 鍚岃鎴栫瓑瑙掔殑琛ヨ鐩哥瓑 4 鍚岃鎴栫瓑瑙掔殑浣欒鐩哥瓑 5 杩囦竴鐐规湁涓斿彧鏈変竴鏉＄洿绾垮拰宸茬煡...
绛旓細tan(A+B)=(tanA+tanB)/(1-tanAtanB) tan(A-B)=(tanA-tanB)/(1+tanAtanB)ctg(A+B)=(ctgActgB-1)/(ctgB+ctgA) ctg(A-B)=(ctgActgB+1)/(ctgB-ctgA) 鍊嶈鍏紡 tan2A=2tanA/(1-tan2A) ctg2A=(ctg2A-1)/2ctga cos2a=cos2a-sin2a=2cos2a-1=1-2sin2a 鍗婅鍏紡 sin(A...
