摘要：两角公式(1)两角和差公式sin(x+y)=sinxcosy+sinycosxsin(x-y)=sinxcosy-sinycosxcos(x+y)=cosxcosy-sinxsinycos(x-y)=cosxcosy+sinxsinytan(x+y)=sin(x+y...

两角公式

(1)两角和差公式

sin(x+y)=sinxcosy+sinycosx

sin(x-y)=sinxcosy-sinycosx

cos(x+y)=cosxcosy-sinxsiny

cos(x-y)=cosxcosy+sinxsiny

tan(x+y)=sin(x+y)/cos(x+y)=sinxcosy+sinycosx/cosxcosy-sinxsiny=tanx+tany/1-tanxtany

tan(x-y)=sin(x-y)/cos(x-y)=sinxcosy-sinycosx/cosxcosy+sinxsiny=tanx-tany/1+tanxtany

证明：单位圆作图

(2)二倍角公式

sin2x=2sinxcosx

推导：sin2x=sin(x+x)=sinxcosx+cosxsinx=2sinxcosx

cos2x=(cosx)²-(sinx)²=2cos²x-1=1-2sin²x (sin²x+cos²x=1)

推导：cos2x=cos(x+x)=cosxcosx-sinxsinx=cos²x-sin²x

tan2x=sin2x/cos2x=2sinxcosx/cos²x-sin²x=2tanx/1-tan²x

三倍角公式

sin3x=sin(2x+x)=sin2xcosx+cos2xsinx=2sinx(1-sin²x)+(1-2sin²x)sinx=3sinx-4sin³x

cos3x=cos(2x+x)=cos2xcosx-sinxsin2x=(2cos²x-1)cosx-2cosx(1-cos²x)=4cos³x-3cosx

tan3x=sin3x/cos3x=tanxtan(π/3+x)tan(π/3-x)

(3)半角公式

sin²(x/2)=(1-cosx)/2

cos²(x/2)=(1+cosx)/2

tan²(x/2)=1-cosx/1+cosx

推导：cosx=2cos²(x/2)-1=1-2sin²(x/2)

(4)辅助角公式

asinx+bcosx=√(a²+b²)[asinx/√(a²+b²)+bcosx/√(a²+b²)]

原理：配凑为sin²m+cos²m的形式，值域为[-√(a²+b²),√(a²+b²)]

(5)两角推诱导例

sin(π+x)=sinπcosx+sinxcosπ=-sinx

cos(π+x)=cosπcosx-sinπsinx=-cosx

sin(π-x)=sinπcosx-sinxcosπ=sinx

cos(π-x)=cosπcosx+sinπsinx=-cosx