三角函数整理

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Sakits 11月 29, 2076

三角函数

诱导公式

sin(2kπ+α)=sinαcos(2kπ+α)=cosαtan(2kπ+α)=tanα\begin{aligned} sin(2k\pi+\alpha)&=sin\alpha\\ cos(2k\pi+\alpha)&=cos\alpha\\ tan(2k\pi+\alpha)&=tan\alpha \end{aligned}

sin(π+α)=sinαcos(π+α)=cosαtan(π+α)=tanα\begin{aligned} sin(\pi+\alpha)&=-sin\alpha\\ cos(\pi+\alpha)&=-cos\alpha\\ tan(\pi+\alpha)&=tan\alpha \end{aligned}

sin(α)=sinαcos(α)=cosαtan(α)=tanα\begin{aligned} sin(-\alpha)&=-sin\alpha\\ cos(-\alpha)&=cos\alpha\\ tan(-\alpha)&=-tan\alpha\\ \end{aligned}

sin(π2+α)=cosαsin(π2α)=cosαcos(π2+α)=sinαcos(π2α)=sinα\begin{aligned} sin(\frac{\pi}{2}+\alpha)&=cos\alpha\\ sin(\frac{\pi}{2}-\alpha)&=cos\alpha\\ cos(\frac{\pi}{2}+\alpha)&=-sin\alpha\\ cos(\frac{\pi}{2}-\alpha)&=sin\alpha \end{aligned}

恒等变换

和差角公式

sin(α+β)=sinαcosβ+cosαsinβsin(αβ)=sinαcosβcosαsinβ\begin{aligned} sin(\alpha + \beta)&=sin\alpha cos\beta+cos\alpha sin\beta\\ sin(\alpha - \beta)&=sin\alpha cos\beta - cos\alpha sin\beta\\ \end{aligned}

cos(α+β)=cosαcosβsinαsinβcos(αβ)=cosαcosβ+sinαsinβ\begin{aligned} cos(\alpha + \beta)&=cos\alpha cos\beta - sin\alpha sin \beta\\ cos(\alpha - \beta)&=cos\alpha cos\beta + sin\alpha sin \beta\\ \end{aligned}

tan(α+β)=tanα+tanβ1tanαtanβtan(αβ)=tanαtanβ1+tanαtanβ\begin{aligned} tan(\alpha + \beta)&=\frac{tan\alpha + tan\beta}{1- tan\alpha tan\beta}\\ tan(\alpha - \beta)&=\frac{tan\alpha - tan\beta}{1 + tan\alpha tan \beta} \end{aligned}

二倍角公式

sin2α=2sinαcosαcos2α=cos2αsin2α=2cos2α1=12sin2αtan2α=2tanα1tan2α\begin{aligned} sin2\alpha&=2sin\alpha cos\alpha\\ \\ cos2\alpha&=cos^2\alpha -sin^2\alpha\\ &=2cos^2\alpha-1\\ &=1-2sin^2\alpha\\ \\ tan2\alpha&=\frac{2tan\alpha}{1-tan^2\alpha} \end{aligned}

降幂公式

cos2α=1+cos2α2sin2α=1cos2α2\begin{aligned} cos^2\alpha=\frac{1+cos2\alpha}{2}\\ \\ sin^2\alpha=\frac{1-cos2\alpha}{2} \end{aligned}

正弦定理

asinA=bsinB=csinC=2R=D\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R=D

余弦定理

c2=a2+b22abcosCc^2=a^2+b^2-2abcosC