学习和计算时特别常用的三角公式

前言

无论搞硬件还是搞软件，数学基础，都很重要，没有一定的数学基础，不管学的语言再多、会的芯片型号再多，也只能算皮毛；做算法的需要数学理论作为支撑，做芯片设计的也需要数理知识作为支撑，总之，对于我们理工科的人，核心的东西还是数学基础，比如三角函数的计算和变换在信号处理中就会经常碰到，有句话常说“代数烦、几何难，三角公式记不完”，三角公式再多，其本质还是通过最基本的公式推导出来的，这里给出常用的三角公式，希望可以帮到大家。

角度与弧度换算

360 ° = 2 π rad 360°=2\pi \;\text{rad} 360°=2πrad

180 ° = π rad 180°=\pi \;\text{rad} 180°=πrad

1 ° = π 180 rad ≈ 0.01745 rad 1°=\frac{\pi}{180}\;\text{rad}\ 0.01745\;\text{rad} 1°=180π​rad≈0.

1 rad = 180 ° π ≈ 57.30 ° 1\;\text{rad}=\frac{180°}{\pi}\ 57.30° 1rad=π180°​≈57.30°

定义式

正弦： sin ⁡ α = a c \text{正弦：}\sin \alpha =\frac{a}{c} 正弦：sinα=ca​

余弦： cos ⁡ α = b c \text{余弦：}\cos \alpha =\frac{b}{c} 余弦：cosα=cb​

正切： tan ⁡ α = a b \text{正切：}\tan \alpha =\frac{a}{b} 正切：tanα=ba​

余切： cot ⁡ α = b a \text{余切：}\cot \alpha =\frac{b}{a} 余切：cotα=ab​

正割： sec ⁡ α = c b \text{正割：}\sec \alpha =\frac{c}{b} 正割：secα=bc​

余割： csc ⁡ α = c a \text{余割：}\csc \alpha =\frac{c}{a} 余割：cscα=ac​

倒数关系：——————— \text{倒数关系：———————} 倒数关系：———————

cot ⁡ α = 1 tan ⁡ α \cot \alpha =\frac{1}{\tan \alpha} cotα=tanα1​

sec ⁡ α = 1 cos ⁡ α \sec \alpha =\frac{1}{\cos \alpha} secα=cosα1​

csc ⁡ α = 1 sin ⁡ α \csc \alpha =\frac{1}{\sin \alpha} cscα=sinα1​

正弦定理

sin ⁡ A a = sin ⁡ B b = sin ⁡ C c = 1 2 R \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R} asinA​=bsinB​=csinC​=2R1​

余弦定理

a 2 = b 2 + c 2 − 2 b c cos ⁡ A a^2=b^2+c^2-2bc\cos A a2=b2+c2−

诱导公式(七组)

sin ⁡ ( 2 k π + α ) = sin ⁡ α k ∈ Z cos ⁡ ( 2 k π + α ) = cos ⁡ α k ∈ Z tan ⁡ ( 2 k π + α ) = tan ⁡ α k ∈ Z cot ⁡ ( 2 k π + α ) = cot ⁡ α k ∈ Z sec ⁡ ( 2 k π + α ) = sec ⁡ α k ∈ Z csc ⁡ ( 2 k π + α ) = csc ⁡ α k ∈ Z ‾ \{\begin{} \sin \left( 2k\pi +\alpha \right) &=\sin \alpha& k\in \{Z}\\ \cos \left( 2k\pi +\alpha \right) &=\cos \alpha& k\in \{Z}\\ \tan \left( 2k\pi +\alpha \right) &=\tan \alpha& k\in \{Z}\\ \cot \left( 2k\pi +\alpha \right) &=\cot \alpha& k\in \{Z}\\ \sec \left( 2k\pi +\alpha \right) &=\sec \alpha& k\in \{Z}\\ \csc \left( 2k\pi +\alpha \right) &=\csc \alpha& k\in \{Z}\\ \end{}} sin(2kπ+α)cos(2kπ+α)tan(2kπ+α)cot(2kπ+α)sec(2kπ+α)csc(2kπ+α)​=sinα=cosα=tanα=cotα=secα=cscα​k∈Zk∈Zk∈Zk∈Zk∈Zk∈Z​​

sin ⁡ ( π + α ) = − sin ⁡ α cos ⁡ ( π + α ) = − cos ⁡ α tan ⁡ ( π + α ) = tan ⁡ α cot ⁡ ( π + α ) = cot ⁡ α sec ⁡ ( π + α ) = − sec ⁡ α csc ⁡ ( π + α ) = − csc ⁡ α ‾ \{\begin{} \sin \left( \pi +\alpha \right) &=-\sin \alpha\\ \cos \left( \pi +\alpha \right) &=-\cos \alpha\\ \tan \left( \pi +\alpha \right) &=\tan \alpha\\ \cot \left( \pi +\alpha \right) &=\cot \alpha\\ \sec \left( \pi +\alpha \right) &=-\sec \alpha\\ \csc \left( \pi +\alpha \right) &=-\csc \alpha\\ \end{}} sin(π+α)cos(π+α)tan(π+α)cot(π+α)sec(π+α)csc(π+α)​=−sinα=−cosα=tanα=cotα=−secα=−cscα​​

sin ⁡ ( − α ) = − sin ⁡ α cos ⁡ ( − α ) = cos ⁡ α tan ⁡ ( − α ) = − tan ⁡ α cot ⁡ ( − α ) = − cot ⁡ α sec ⁡ ( − α ) = sec ⁡ α csc ⁡ ( − α ) = − csc ⁡ α ‾ \{\begin{} \sin \left( -\alpha \right) &=-\sin \alpha\\ \cos \left( -\alpha \right) &=\cos \alpha\\ \tan \left( -\alpha \right) &=-\tan \alpha\\ \cot \left( -\alpha \right) &=-\cot \alpha\\ \sec \left( -\alpha \right) &=\sec \alpha\\ \csc \left( -\alpha \right) &=-\csc \alpha\\ \end{}} sin(−α)cos(−α)tan(−α)cot(−α)sec(−α)csc(−α)​=−sinα=cosα=−tanα=−cotα=secα=−cscα​​

sin ⁡ ( π − α ) = sin ⁡ α cos ⁡ ( π − α ) = − cos ⁡ α tan ⁡ ( π − α ) = − tan ⁡ α cot ⁡ ( π − α ) = − cot ⁡ α sec ⁡ ( π − α ) = − sec ⁡ α csc ⁡ ( π − α ) = csc ⁡ α ‾ \{\begin{} \sin \left( \pi -\alpha \right) &=\sin \alpha\\ \cos \left( \pi -\alpha \right) &=-\cos \alpha\\ \tan \left( \pi -\alpha \right) &=-\tan \alpha\\ \cot \left( \pi -\alpha \right) &=-\cot \alpha\\ \sec \left( \pi -\alpha \right) &=-\sec \alpha\\ \csc \left( \pi -\alpha \right) &=\csc \alpha\\ \end{}} sin(π−α)cos(π−α)tan(π−α)cot(π−α)sec(π−α)csc(π−α)​=sinα=−cosα=−tanα=−cotα=−secα=cscα​​

sin ⁡ ( 2 π − α ) = − sin ⁡ α cos ⁡ ( 2 π − α ) = cos ⁡ α tan ⁡ ( 2 π − α ) = − tan ⁡ α cot ⁡ ( 2 π − α ) = − cot ⁡ α sec ⁡ ( 2 π − α ) = sec ⁡ α csc ⁡ ( 2 π − α ) = − csc ⁡ α ‾ \{\begin{} \sin \left( 2\pi -\alpha \right) &=-\sin \alpha\\ \cos \left( 2\pi -\alpha \right) &=\cos \alpha\\ \tan \left( 2\pi -\alpha \right) &=-\tan \alpha\\ \cot \left( 2\pi -\alpha \right) &=-\cot \alpha\\ \sec \left( 2\pi -\alpha \right) &=\sec \alpha\\ \csc \left( 2\pi -\alpha \right) &=-\csc \alpha\\ \end{}} sin(2π−α)cos(2π−α)tan(2π−α)cot(2π−α)sec(2π−α)csc(2π−α)​=−sinα=cosα=−tanα=−cotα=secα=−cscα​​

sin ⁡ ( π 2 + α ) = cos ⁡ α cos ⁡ ( π 2 + α ) = − sin ⁡ α tan ⁡ ( π 2 + α ) = − cot ⁡ α cot ⁡ ( π 2 + α ) = − tan ⁡ α sec ⁡ ( π 2 + α ) = − csc ⁡ α csc ⁡ ( π 2 + α ) = sec ⁡ α ‾ \{\begin{} \sin \left( \frac{\pi}{2}+\alpha \right) &=\cos \alpha\\ \cos \left( \frac{\pi}{2}+\alpha \right) &=-\sin \alpha\\ \tan \left( \frac{\pi}{2}+\alpha \right) &=-\cot \alpha\\ \cot \left( \frac{\pi}{2}+\alpha \right) &=-\tan \alpha\\ \sec \left( \frac{\pi}{2}+\alpha \right) &=-\csc \alpha\\ \csc \left( \frac{\pi}{2}+\alpha \right) &=\sec \alpha\\ \end{}} sin(2π​+α)cos(2π​+α)tan(2π​+α)cot(2π​+α)sec(2π​+α)csc(2π​+α)​=cosα=−sinα=−cotα=−tanα=−cscα=secα​​

sin ⁡ ( π 2 − α ) = cos ⁡ α cos ⁡ ( π 2 − α ) = sin ⁡ α tan ⁡ ( π 2 − α ) = cot ⁡ α cot ⁡ ( π 2 − α ) = tan ⁡ α csc ⁡ ( π 2 − α ) = sec ⁡ α sec ⁡ ( π 2 − α ) = csc ⁡ α ‾ \{\begin{} \sin \left( \frac{\pi}{2}-\alpha \right) &=\cos \alpha\\ \cos \left( \frac{\pi}{2}-\alpha \right) &=\sin \alpha\\ \tan \left( \frac{\pi}{2}-\alpha \right) &=\cot \alpha\\ \cot \left( \frac{\pi}{2}-\alpha \right) &=\tan \alpha\\ \csc \left( \frac{\pi}{2}-\alpha \right) &=\sec \alpha\\ \sec \left( \frac{\pi}{2}-\alpha \right) &=\csc \alpha\\ \end{}} sin(2π​−α)cos(2π​−α)tan(2π​−α)cot(2π​−α)csc(2π​−α)sec(2π​−α)​=cosα=sinα=cotα=tanα=secα=cscα​​

两角和公式(加法公式)[三组]

sin ⁡ ( α + β ) = sin ⁡ α cos ⁡ β + cos ⁡ α sin ⁡ β \sin \left( \alpha +\beta \right) =\sin \alpha \cos \beta +\cos \alpha \sin \beta sin(α+β)=sinαcosβ+cosαsinβ

sin ⁡ ( α − β ) = sin ⁡ α cos ⁡ β − cos ⁡ α sin ⁡ β \sin \left( \alpha -\beta \right) =\sin \alpha \cos \beta -\cos \alpha \sin \beta sin(α−β)=sinαcosβ−cosαsinβ

cos ⁡ ( α + β ) = cos ⁡ α cos ⁡ β − sin ⁡ α sin ⁡ β \cos \left( \alpha +\beta \right) =\cos \alpha \cos \beta -\sin \alpha \sin \beta cos(α+β)=cosαcosβ−sinαsinβ

cos ⁡ ( α − β ) = cos ⁡ α cos ⁡ β + sin ⁡ α sin ⁡ β \cos \left( \alpha -\beta \right) =\cos \alpha \cos \beta +\sin \alpha \sin \beta cos(α−β)=cosαcosβ+sinαsinβ

tan ⁡ ( α + β ) = tan ⁡ α + tan ⁡ β 1 − tan ⁡ α tan ⁡ β \tan \left( \alpha +\beta \right) =\frac{\tan \alpha +\tan \beta}{1-\tan \alpha \tan \beta} tan(α+β)=1−tanαtanβtanα+tanβ​

tan ⁡ ( α − β ) = tan ⁡ α − tan ⁡ β 1 + tan ⁡ α tan ⁡ β \tan \left( \alpha -\beta \right) =\frac{\tan \alpha -\tan \beta}{1+\tan \alpha \tan \beta} tan(α−β)=1+tanαtanβtanα−tanβ​

倍角公式

sin ⁡ 2 α = 2 sin ⁡ α cos ⁡ α \sin 2\alpha =2\sin \alpha \cos \alpha sin2α=2sinαcosα

cos ⁡ 2 α = cos ⁡ 2 α − sin ⁡ 2 α \cos 2\alpha =\cos ^2\alpha -\sin ^2\alpha cos2α=cos2α−sin2α

= 2 cos ⁡ 2 α − 1 =2\cos ^2\alpha -1 =2cos2α−1

= 1 − 2 sin ⁡ 2 α =1-2\sin ^2\alpha =1−2sin2α

tan ⁡ 2 α = 2 tan ⁡ α 1 − tan ⁡ 2 α \tan 2\alpha =\frac{2\tan \alpha}{1-\tan ^2\alpha} tan2α=1−tan2α2tanα​

三倍角公式

sin ⁡ 3 α = 3 sin ⁡ α − 4 sin ⁡ 3 α \sin 3\alpha =3\sin \alpha -4\sin ^3\alpha sin3α=3sinα−4sin3α

cos ⁡ 3 α = 4 cos ⁡ 3 α − 3 cos ⁡ α \cos 3\alpha =4\cos ^3\alpha -3\cos \alpha cos3α=4cos3α−3cosα

tan ⁡ 3 α = tan ⁡ α tan ⁡ ( π 3 + α ) tan ⁡ ( π 3 − α ) \tan 3\alpha =\tan \alpha \tan \left( \frac{\pi}{3}+\alpha \right) \tan \left( \frac{\pi}{3}-\alpha \right) tan3α=tanαtan(3π​+α)tan(3π​−α)

半角公式

sin ⁡ 2 α 2 = 1 − cos ⁡ α 2 \sin ^2\frac{\alpha}{2}=\frac{1-\cos \alpha}{2} sin22α​=21−cosα​

cos ⁡ 2 α 2 = 1 + cos ⁡ α 2 \cos ^2\frac{\alpha}{2}=\frac{1+\cos \alpha}{2} cos22α​=21+cosα​

tan ⁡ α 2 = sin ⁡ α 1 + cos ⁡ α \tan \frac{\alpha}{2}=\frac{\sin \alpha}{1+\cos \alpha} tan2α​=1+cosαsinα​

和差化积

sin ⁡ α + sin ⁡ β = 2 sin ⁡ α + β 2 ⋅ cos ⁡ α − β 2 \sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta}{2}\cdot \cos \frac{\alpha -\beta}{2} sinα+sinβ=2sin2α+β​⋅cos2α−β​

sin ⁡ α − sin ⁡ β = 2 sin ⁡ α − β 2 ⋅ cos ⁡ α + β 2 \sin \alpha -\sin \beta =2\sin \frac{\alpha -\beta}{2}\cdot \cos \frac{\alpha +\beta}{2} sinα−sinβ=2sin2α−β​⋅cos2α+β​

cos ⁡ α + cos ⁡ β = 2 cos ⁡ α + β 2 ⋅ cos ⁡ α − β 2 \cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta}{2}\cdot \cos \frac{\alpha -\beta}{2} cosα+cosβ=2cos2α+β​⋅cos2α−β​

cos ⁡ α − cos ⁡ β = − 2 sin ⁡ α + β 2 ⋅ sin ⁡ α − β 2 \cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta}{2}\cdot \sin \frac{\alpha -\beta}{2} cosα−cosβ=−2sin2α+β​⋅sin2α−β​

积化和差

2 cos ⁡ α cos ⁡ β = cos ⁡ ( α − β ) + cos ⁡ ( α + β ) 2\cos \alpha \cos \beta =\cos \left( \alpha -\beta \right) +\cos \left( \alpha +\beta \right) 2cosαcosβ=cos(α−β)+cos(α+β)

2 sin ⁡ α sin ⁡ β = cos ⁡ ( α + β ) − cos ⁡ ( α + β ) 2\sin \alpha \sin \beta =\cos \left( \alpha +\beta \right) -\cos \left( \alpha +\beta \right) 2sinαsinβ=cos(α+β)−cos(α+β)

2 sin ⁡ α cos ⁡ β = sin ⁡ ( α − β ) + sin ⁡ ( α + β ) 2\sin \alpha \cos \beta =\sin \left( \alpha -\beta \right) +\sin \left( \alpha +\beta \right) 2sinαcosβ=sin(α−β)+sin(α+β)

万能公式（毕达哥拉斯恒等式）

第一恒等式： sin ⁡ 2 α + cos ⁡ 2 α = 1 \text{第一恒等式：}\sin ^2\alpha +\cos ^2\alpha =1 第一恒等式：sin2α+cos2α=1

第二恒等式： tan ⁡ 2 α + 1 = sec ⁡ 2 α \text{第二恒等式：}\tan ^2\alpha +1=\sec ^2\alpha 第二恒等式：tan2α+1=sec2α

第三恒等式： cot ⁡ 2 α + 1 = csc ⁡ 2 α \text{第三恒等式：}\cot ^2\alpha +1=\csc ^2\alpha 第三恒等式：cot2α+1=csc2α