和角公式

$$ \large \begin{align*} & \sin(\alpha \pm \beta)=\sin\alpha\cos\beta \pm \cos\alpha\sin\beta\\[2mm] & \cos(\alpha \pm \beta)=\cos\alpha\cos\beta \mp \sin\alpha\sin\beta\\[2mm] & \tan(\alpha \pm \beta)=\frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}\\[4mm] & \cot(\alpha \pm \beta)=\frac{\cot\alpha\cot\beta \mp 1}{\cot\beta \pm \cot\alpha} \end{align*} $$

降幂公式

$$ \large \begin{align*} & \sin^2\alpha=\frac{1}{2}\left(1-\cos 2\alpha\right)\\[4mm] & \cos^2\alpha=\frac{1}{2}\left(1+\cos 2\alpha\right)\\[4mm] & \sin^3\alpha=\frac{1}{4}(3\sin\alpha-\sin 3\alpha)\\[4mm] & \cos^3\alpha=\frac{1}{4}(3\cos\alpha+\cos 3\alpha)\\[4mm] & \sin^4\alpha=\frac{1}{8}(\cos 4\alpha-4\cos 2\alpha+3)\\[4mm] & \cos^4\alpha=\frac{1}{8}(\cos 4\alpha+4\cos 2\alpha+3)\\[4mm] & \sin^{2n}\alpha=\frac{1}{2^{2n-1}}\left[\sum_{i=0}^{n-1}(-1)^{n+i}C_{2n}^{i}\cos(2n-2i)\alpha+\frac{1}{2}C_{2n}^{n}\right]\\[6mm] & \cos^{2n}\alpha=\frac{1}{2^{2n-1}}\left[\sum_{i=0}^{n-1}C_{2n}^{i}\cos(2n-2i)\alpha+\frac{1}{2}C_{2n}^{n}\right]\\[6mm] & \sin^{2n+1}\alpha=\frac{1}{2^{2n}}\sum_{i=0}^{n}(-1)^{n+i}C_{2n+1}^{i}\sin(2n-2i+1)\alpha\\[6mm] & \cos^{2n+1}\alpha=\frac{1}{2^{2n}}\sum_{i=0}^{n}C_{2n+1}^{i}\cos(2n-2i+1)\alpha \end{align*} $$

倍角公式

$$ \large \begin{align*} & \sin 2\alpha = 2\sin\alpha\cos\alpha\\[2mm] & \cos 2\alpha = 2\cos^2\alpha-1=1-2\sin^2\alpha=\cos^2\alpha-\sin^2\alpha\\[2mm] & \tan 2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}\\[4mm] & \sin 3\alpha=3\sin\alpha-4\sin^3\alpha\\[2mm] & \cos 3\alpha=4\cos^3\alpha-3\cos\alpha\\[2mm] & \tan 3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha}\\[4mm] & \sin n\alpha=\left\{\begin{aligned} & C_n^1\sin\alpha\cos^{n-1}\alpha-C_n^3\sin^3\alpha\cos^{n-3}\alpha+\cdots+(-1)^{\frac{n-1}{2}}\sin^n\alpha\quad & (n\text{是奇数})\\[2mm] & C_n^1\sin\alpha\cos^{n-1}\alpha-C_n^3\sin^3\alpha\cos^{n-3}\alpha+\cdots+(-1)^{\frac{n-2}{2}}\cdot n\sin^{n-1}\alpha\cos\alpha\quad & (n\text{是偶数}) \end{aligned}\right.\\[2mm] & \cos n\alpha=\left\{\begin{aligned} & \cos^n\alpha-C_n^2\sin^2\alpha\cos^{n-2}\alpha+\cdots+(-1)^{\frac{n-1}{2}}\cdot n\sin^{n-1}\alpha\cos\alpha\quad & (n\text{是奇数})\\[2mm] & \cos^n\alpha-C_n^2\sin^2\alpha\cos^{n-2}\alpha+\cdots+(-1)^{\frac{n}{2}}\cdot \sin^{n}\alpha\quad & (n\text{是偶数}) \end{aligned}\right. \end{align*} $$

半角公式

$$ \large \begin{align*} & \sin\frac{\alpha}{2}=\pm \sqrt{\frac{1-\cos\alpha}{2}}\\[4mm] & \cos\frac{\alpha}{2}=\pm \sqrt{\frac{1+\cos\alpha}{2}}\\[4mm] & \tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha} \end{align*} $$

和差化积

$$ \large \begin{align*} & \sin\alpha \pm \sin\beta=2\sin\frac{\alpha \pm \beta}{2}\cos\frac{\alpha \mp \beta}{2}\\[4mm] & \cos\alpha + \cos\beta=2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2}\\[4mm] & \cos\alpha - \cos\beta=2\sin\frac{\alpha + \beta}{2}\sin\frac{\beta - \alpha}{2} \end{align*} $$

积化和差

$$ \large \begin{align*} & \sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)]\\[4mm] & \cos\alpha\sin\beta=\frac{1}{2}[\sin(\alpha+\beta)-\sin(\alpha-\beta)]\\[4mm] & \cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha-\beta)]\\[4mm] & \sin\alpha\sin\beta=\frac{1}{2}[\cos(\alpha-\beta)-\cos(\alpha+\beta)] \end{align*} $$

万能公式

$$ \large \begin{align*} & \sin\alpha=\frac{2\tan\dfrac{\alpha}{2}}{1+\tan^2\dfrac{\alpha}{2}}\\[8mm] & \cos\alpha=\frac{1-\tan^2\dfrac{\alpha}{2}}{1+\tan^2\dfrac{\alpha}{2}}\\[8mm] & \tan\alpha=\frac{2\tan\dfrac{\alpha}{2}}{1-\tan^2\dfrac{\alpha}{2}}\\[8mm] & \cot\alpha=\frac{1-\tan^2\dfrac{\alpha}{2}}{2\tan\dfrac{\alpha}{2}}\\[8mm] & \sec\alpha=\frac{1+\tan^2\dfrac{\alpha}{2}}{1-\tan^2\dfrac{\alpha}{2}}\\[8mm] & \csc\alpha=\frac{1+\tan^2\dfrac{\alpha}{2}}{2\tan\dfrac{\alpha}{2}}\\[8mm] \end{align*} $$