三角函数常用公式

基本公式

$$ \begin{aligned} & \sin ^2 \alpha+\cos ^2 \alpha=1 \\ & 1+\cot ^2 \alpha=\csc ^2 \alpha \\ & \frac{\sin \alpha}{\cos \alpha}=\tan \alpha \\ & \tan \alpha \cdot \cot \alpha=1 \\ & \cos \alpha \cdot \sec \alpha=1 \\ & \sin \alpha \cdot \csc \alpha=1 \end{aligned} $$

倍角公式

$$ \begin{aligned} \sin 2 \alpha & =2 \sin \alpha \cos \alpha \\ \cos 2 \alpha & =\cos ^2 \alpha-\sin ^2 \alpha \\ & =2 \cos ^2 \alpha-1 \\ & =1-2 \sin ^2 \alpha \\ \tan 2 \alpha & =\frac{2 \tan \alpha}{1-\tan ^2 \alpha} \end{aligned} $$

正弦定理

$$ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R $$

余弦定理

$$ \begin{aligned} & a^2=b^2+c^2-2 b c \cos A \\ & \cos A=\frac{b^2+c^2-a^2}{2 b c} \end{aligned} $$

和差公式

$$ \begin{aligned} & \sin (\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ & \cos (\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ & \tan (\alpha \pm \beta)=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} \end{aligned} $$

万能公式

$$ \begin{aligned} \sin 2 \alpha & =\frac{2 \tan \alpha}{1+\tan ^2 \alpha} \\ \cos 2 \alpha & =\frac{1-\tan ^2 \alpha}{1+\tan ^2 \alpha} \\ \tan 2 \alpha & =\frac{2 \tan \alpha}{1-\tan ^2 \alpha} \end{aligned} $$

积化和差

$$ \begin{aligned} \sin \alpha \cos \beta & =\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)] \\ \cos \alpha \sin \beta & =\frac{1}{2}[\sin (\alpha+\beta)-\sin (\alpha-\beta)] \\ \cos \alpha \cos \beta & =\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)] \\ \sin \alpha \sin \beta & =-\frac{1}{2}[\cos (\alpha+\beta)-\cos (\alpha-\beta)] \end{aligned} $$

和差化积

$$ \begin{aligned} & \sin \alpha+\sin \beta=2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \\ & \sin \alpha-\sin \beta=2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \\ & \cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \\ & \cos \alpha-\cos \beta=-2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \end{aligned} $$

特殊角

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text { 角 } \alpha & 0^{\circ} & 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} & 120^{\circ} & 150^{\circ} & 180^{\circ} & 270^{\circ} \\ \hline \text { 弧度制 } & 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} & \frac{2 \pi}{3} & \frac{5 \pi}{6} & \pi & \frac{3 \pi}{2} \\ \hline \sin \alpha & 0 & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & 1 & \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 & -1 \\ \hline \cos \alpha & 1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{1}{2} & 0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2} & -1 & 0 \\ \hline \tan \alpha & 0 & \frac{\sqrt{3}}{3} & 1 & \sqrt{3} & \text { 不存在 } & -\sqrt{3} & -\frac{\sqrt{3}}{3} & 0 & \text { 不存在 } \\ \hline \end{array} $$