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三角函数汇总
三角函数角度转换关系
- 奇变偶不变，符号看象限
| 正弦函数 | 余弦函数 | 正切函数 |
|---|---|---|
| sin ( − α ) = − sin α \sin(-\alpha) = -\sin\alpha sin(−α)=−sinα | cos ( − α ) = cos α \cos(-\alpha) = \cos \alpha cos(−α)=cosα | tan ( − α ) = − tan α \tan(-\alpha) = -\tan\alpha tan(−α)=−tanα |
| sin ( π 2 + α ) = cos α \sin\Big(\dfrac \pi2 + \alpha \Big) = \cos \alpha sin(2π+α)=cosα | cos ( π 2 + α ) = − sin α \cos\Big(\dfrac \pi2 + \alpha \Big) = -\sin \alpha cos(2π+α)=−sinα | tan ( π 2 + α ) = − cot α \tan\Big(\dfrac \pi2 + \alpha \Big) = -\cot \alpha tan(2π+α)=−cotα |
| sin ( π 2 − α ) = cos α \sin\Big(\dfrac \pi2 – \alpha \Big) = \cos \alpha sin(2π−α)=cosα | cos ( π 2 − α ) = sin α \cos\Big(\dfrac \pi2 – \alpha \Big) = \sin \alpha cos(2π−α)=sinα | tan ( π 2 − α ) = cot α \tan\Big(\dfrac \pi2 – \alpha \Big) = \cot \alpha tan(2π−α)=cotα |
| sin ( π + α ) = − sin α \sin\Big(\pi + \alpha \Big) = -\sin \alpha sin(π+α)=−sinα | cos ( π + α ) = − cos α \cos\Big(\pi + \alpha \Big) = -\cos \alpha cos(π+α)=−cosα | tan ( π + α ) = tan α \tan\Big(\pi + \alpha \Big) = \tan \alpha tan(π+α)=tanα |
| sin ( π − α ) = sin α \sin\Big(\pi – \alpha \Big) = \sin \alpha sin(π−α)=sinα | cos ( π − α ) = − cos α \cos\Big(\pi – \alpha \Big) = -\cos \alpha cos(π−α)=−cosα | tan ( π − α ) = − tan α \tan\Big(\pi – \alpha \Big) = -\tan \alpha tan(π−α)=−tanα |
| sin ( π + α ) = − sin α \sin\Big(\pi + \alpha \Big) = -\sin \alpha sin(π+α)=−sinα | cos ( π + α ) = − cos α \cos\Big(\pi + \alpha \Big) = -\cos \alpha cos(π+α)=−cosα | tan ( π + α ) = tan α \tan\Big(\pi + \alpha \Big) = \tan \alpha tan(π+α)=tanα |
| sin ( 2 π − α ) = − sin α \sin\Big(2\pi – \alpha \Big) = -\sin \alpha sin(2π−α)=−sinα | cos ( 2 π − α ) = cos α \cos\Big(2\pi – \alpha \Big) = \cos \alpha cos(2π−α)=cosα | tan ( 2 π − α ) = − tan α \tan\Big(2\pi – \alpha \Big) = -\tan \alpha tan(2π−α)=−tanα |
倍角公式
- 正弦倍角公式
sin 2 α = 2 sin α cos α = 2 tan α 1 + tan 2 α \begin{split} \sin 2\alpha &= 2\sin \alpha \cos \alpha \\ &= \dfrac{2\tan \alpha}{1 + \tan^2 \alpha} \\ \end{split} sin2α=2sinαcosα=1+tan2α2tanα - 余弦倍角公式
cos 2 α = cos 2 α − sin 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α = 1 − tan 2 α 1 + tan 2 α \begin{split} \cos 2\alpha &= \cos^2 \alpha – \sin^2 \alpha \\ & = 2\cos^2 \alpha – 1 \\ &= 1 – 2\sin^2 \alpha \\ &= \dfrac{1 – \tan^2 \alpha}{1 + \tan^2 \alpha} \\ \end{split} cos2α=cos2α−sin2α=2cos2α−1=1−2sin2α=1+tan2α1−tan2α - 正切倍角公式
tan 2 α = 2 tan α 1 − tan 2 α \tan 2\alpha = \dfrac{2\tan \alpha}{1 – \tan^2 \alpha} tan2α=1−tan2α2tanα
半角公式
sin 2 α = 1 − cos 2 α 2 cos 2 α = 1 + cos 2 α 2 \begin{split} \sin^2 \alpha &= \dfrac{1 – \cos 2\alpha}{2} \\ \cos^2 \alpha &= \dfrac{1 + \cos 2\alpha}{2} \\ \end{split} sin2αcos2α=21−cos2α=21+cos2α
同角三角函数
s i n 2 α + c o s 2 α = 1 1 + t a n 2 α = s e c 2 α 1 + c o t 2 α = c s c 2 α sin^2\alpha + cos^2\alpha = 1 \\ 1 + tan^2\alpha = sec^2\alpha \\ 1 + cot^2 \alpha = csc^2 \alpha sin2α+cos2α=11+tan2α=sec2α1+cot2α=csc2α
积化和差三角函数
s i n α s i n β = − 1 2 [ c o s ( α + β ) − c o s ( α − β ) ] s i n α c o s β = 1 2 [ s i n ( α + β ) + s i n ( α − β ) ] c o s α c o s β = 1 2 [ c o s ( α + β ) + c o s ( α − β ) ] sin\alpha sin\beta = -\frac 12 \Big[ cos(\alpha + \beta) – cos(\alpha – \beta)\Big] \\ sin\alpha cos\beta = \frac 12 \Big[ sin(\alpha + \beta) + sin(\alpha – \beta)\Big] \\ cos\alpha cos\beta = \frac 12 \Big[ cos(\alpha + \beta) + cos(\alpha – \beta)\Big] \\ sinαsinβ=−21[cos(α+β)−cos(α−β)]sinαcosβ=21[sin(α+β)+sin(α−β)]cosαcosβ=21[cos(α+β)+cos(α−β)]
和差化积三角函数
s i n α + s i n β = 2 s i n α + β 2 c o s α − β 2 c o s α + c o s β = 2 c o s α + β 2 c o s α − β 2 c o s α − c o s β = − 2 s i n α + β 2 s i n α − β 2 a s i n α + b c o s α = a 2 + b 2 s i n ( α + ψ ) sin\alpha + sin\beta = 2sin\frac{\alpha + \beta}{2}cos\frac{\alpha – \beta}{2} \\ cos\alpha + cos\beta = 2cos\frac{\alpha + \beta}{2}cos\frac{\alpha – \beta}{2} \\ cos\alpha – cos\beta = -2sin\frac{\alpha + \beta}{2}sin\frac{\alpha – \beta}{2} \\ asin\alpha + bcos\alpha = \sqrt{a^2 + b^2}sin(\alpha + \psi) sinα+sinβ=2sin2α+βcos2α−βcosα+cosβ=2cos2α+βcos2α−βcosα−cosβ=−2sin2α+βsin2α−βasinα+bcosα=a2+b2sin(α+ψ)
倍角公式
- 正弦倍角公式
sin 2 α = 2 sin α cos α = 2 tan α 1 + tan 2 α \begin{split} \sin 2\alpha &= 2\sin \alpha \cos \alpha \\ &= \dfrac{2\tan \alpha}{1 + \tan^2 \alpha} \\ \end{split} sin2α=2sinαcosα=1+tan2α2tanα - 余弦倍角公式
cos 2 α = cos 2 α − sin 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α = 1 − tan 2 α 1 + tan 2 α \begin{split} \cos 2\alpha &= \cos^2 \alpha – \sin^2 \alpha \\ & = 2\cos^2 \alpha – 1 \\ &= 1 – 2\sin^2 \alpha \\ &= \dfrac{1 – \tan^2 \alpha}{1 + \tan^2 \alpha} \\ \end{split} cos2α=cos2α−sin2α=2cos2α−1=1−2sin2α=1+tan2α1−tan2α - 正切倍角公式
tan 2 α = 2 tan α 1 − tan 2 α \tan 2\alpha = \dfrac{2\tan \alpha}{1 – \tan^2 \alpha} tan2α=1−tan2α2tanα
半角公式
sin 2 α = 1 − cos 2 α 2 cos 2 α = 1 + cos 2 α 2 \begin{split} \sin^2 \alpha &= \dfrac{1 – \cos 2\alpha}{2} \\ \cos^2 \alpha &= \dfrac{1 + \cos 2\alpha}{2} \\ \end{split} sin2αcos2α=21−cos2α=21+cos2α
同角三角函数
s i n 2 α + c o s 2 α = 1 1 + t a n 2 α = s e c 2 α 1 + c o t 2 α = c s c 2 α sin^2\alpha + cos^2\alpha = 1 \\ 1 + tan^2\alpha = sec^2\alpha \\ 1 + cot^2 \alpha = csc^2 \alpha sin2α+cos2α=11+tan2α=sec2α1+cot2α=csc2α
积化和差三角函数
s i n α s i n β = − 1 2 [ c o s ( α + β ) − c o s ( α − β ) ] s i n α c o s β = 1 2 [ s i n ( α + β ) + s i n ( α − β ) ] c o s α c o s β = 1 2 [ c o s ( α + β ) + c o s ( α − β ) ] sin\alpha sin\beta = -\frac 12 \Big[ cos(\alpha + \beta) – cos(\alpha – \beta)\Big] \\ sin\alpha cos\beta = \frac 12 \Big[ sin(\alpha + \beta) + sin(\alpha – \beta)\Big] \\ cos\alpha cos\beta = \frac 12 \Big[ cos(\alpha + \beta) + cos(\alpha – \beta)\Big] \\ sinαsinβ=−21[cos(α+β)−cos(α−β)]sinαcosβ=21[sin(α+β)+sin(α−β)]cosαcosβ=21[cos(α+β)+cos(α−β)]
和差化积三角函数
s i n α + s i n β = 2 s i n α + β 2 c o s α − β 2 c o s α + c o s β = 2 c o s α + β 2 c o s α − β 2 c o s α − c o s β = − 2 s i n α + β 2 s i n α − β 2 a s i n α + b c o s α = a 2 + b 2 s i n ( α + ψ ) sin\alpha + sin\beta = 2sin\frac{\alpha + \beta}{2}cos\frac{\alpha – \beta}{2} \\ cos\alpha + cos\beta = 2cos\frac{\alpha + \beta}{2}cos\frac{\alpha – \beta}{2} \\ cos\alpha – cos\beta = -2sin\frac{\alpha + \beta}{2}sin\frac{\alpha – \beta}{2} \\ asin\alpha + bcos\alpha = \sqrt{a^2 + b^2}sin(\alpha + \psi) sinα+sinβ=2sin2α+βcos2α−βcosα+cosβ=2cos2α+βcos2α−βcosα−cosβ=−2sin2α+βsin2α−βasinα+bcosα=a2+b2sin(α+ψ)
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