三角函数常用公式总结
下文所有与开根结果有关的 \(\pm\) 号均与角 \(\alpha\) 有关。
一、基本公式
1. 诱导公式
\[ \begin{aligned} &\sin ^{2} \alpha+\cos ^{2} \alpha=1 &&&& \sin (\alpha+2 \pi)=\sin \alpha &&&& \sin (\pi+\alpha)=-\sin \alpha \\[6pt]&\tan \alpha=\dfrac{\sin \alpha}{\cos \alpha} &&&&\cos (\alpha+2 \pi)=\cos \alpha &&&& \cos (\pi+\alpha)=-\cos \alpha \\[6pt]&\tan\alpha=\dfrac{1}{\cot \alpha} &&&& \tan (\alpha+2 \pi)=\tan \alpha &&&& \tan (\pi+\alpha)=\tan \alpha \\[6pt] \\[6pt]&\sin (-\alpha)=-\sin \alpha &&&& \sin (\pi-\alpha)=\sin \alpha &&&& \sin \left(\dfrac{\pi}{2}-\alpha\right)=\cos \alpha \\[6pt]&\cos (-\alpha)=\cos \alpha &&&& \cos (\pi-\alpha)=-\cos \alpha &&&& \cos \left(\dfrac{\pi}{2}-\alpha\right)=\sin \alpha \\[6pt]&\tan (-\alpha)=-\tan \alpha &&&& \tan (\pi -\alpha)=-\tan \alpha &&&& \tan \left(\frac{\pi}{2}-\alpha\right) = \cot \alpha \\[6pt] \\[6pt]&\sin \left(\dfrac{\pi}{2}+\alpha\right)=\cos \alpha &&&& \sin\left(\dfrac{3\pi}{2}+\alpha\right)=-\cos\alpha &&&& \sin\left(\dfrac{3\pi}{2}-\alpha\right)=-\cos\alpha \\[6pt]&\cos \left(\dfrac{\pi}{2}+\alpha\right)=-\sin \alpha &&&& \cos\left(\dfrac{3\pi}{2}+\alpha\right)=\sin\alpha &&&& \cos\left(\dfrac{3\pi}{2}-\alpha\right)=-\sin\alpha \\[6pt]&\tan \left(\dfrac{\pi}{2}+\alpha\right) = -\cot \alpha &&&& \tan\left(\dfrac{3\pi}{2}+\alpha\right)=-\cot\alpha &&&& \tan\left(\dfrac{3\pi}{2}-\alpha\right)=\cot\alpha \end{aligned} \]
常用结论：
奇变偶不变，符号看象限： \[ \sin\left(\dfrac{k\pi}{2}\pm \alpha\right),k\in\mathbb{Z} \]
一些
排版不好看但常用的公式 \[ \sin \alpha=\pm \sqrt{1-\cos ^2 \alpha} \\[6pt] \cos \alpha=\pm \sqrt{1-\sin ^2 \alpha} \\[6pt] (\sin \alpha+\cos \alpha)^2+(\sin \alpha-\cos \alpha)^2=2 \\[6pt] (\sin \alpha \pm \cos \alpha)^2=1 \pm \sin 2 \alpha \]
2. 和差角公式
\[ \begin{aligned} & \sin (\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\[6pt]& \cos (\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\[6pt]& \tan (\alpha \pm \beta)=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} \end{aligned} \]
3. 辅助角公式
这是一个比较有用的公式，在考试的时候一般给的 \(\frac{b}{a}\) 都很好求 \(\arctan\) \[ a \sin x+b \cos x=\sqrt{a^{2}+b^{2}} \sin \left(x+\arctan \dfrac{b}{a}\right) \] 辅助角公式常用结论： \[ \begin{aligned} & \sin x \pm \cos x=\sqrt{2} \sin \left(x \pm \frac{\pi}{4}\right) \\[6pt]& \cos x \pm \sin x=\sqrt{2} \cos \left(x \mp \frac{\pi}{4}\right) \end{aligned} \]
二、推广公式
1. 二倍角公式
\[ \begin{aligned} & \sin 2 \alpha=2 \sin \alpha \cos \alpha \\[6pt]& \cos 2 \alpha=\cos ^2 \alpha-\sin ^2 \alpha \\[6pt]& \tan 2 \alpha=\frac{2 \tan \alpha}{1-\tan ^2 \alpha} \end{aligned} \]
特别地，\(\cos 2\alpha = 1-2 \sin ^{2} \alpha=2 \cos ^{2} \alpha-1\) 。
2. 半角公式
\[ \begin{aligned} & \sin \frac{\alpha}{2}=\pm \sqrt{\frac{1-\cos \alpha}{2}} \\[3pt] & \cos \frac{\alpha}{2}=\pm \sqrt{\frac{1+\cos \alpha}{2}} \\[3pt] & \tan \frac{\alpha}{2}=\pm \sqrt{\frac{1-\cos \alpha}{1+\cos \alpha}}= \dfrac{1-\cos\alpha}{\sin\alpha} \end{aligned} \]
2.1 推广
$$ \[\begin{aligned} &\sin ^{2} \alpha=\dfrac{1-\cos 2 \alpha}{2} && && &&1+\cos \alpha=2 \cos ^{2} \dfrac{\alpha}{2} \\[3pt]&\cos ^{2} \alpha=\dfrac{1+\cos 2 \alpha}{2} && && && 1-\cos\alpha = 2\sin^2\dfrac{\alpha}{2} \end{aligned}\]$$
3. 积化和差公式
\[ \begin{aligned} & \sin x \cos y=\frac{1}{2}[\sin (x+y)+\sin (x-y)] \\[3pt] & \cos x \sin y=\frac{1}{2}[\sin (x+y)-\sin (x-y)] \\[3pt] & \cos x \cos y=\frac{1}{2}[\cos (x+y)+\cos (x-y)] \\[3pt] & \sin x \sin y=-\frac{1}{2}[\cos (x+y)-\cos (x-y)] \end{aligned} \]
4. 和差化积公式
\[ \begin{aligned} & \sin x+\sin y=2 \sin \frac{x+y}{2} \cos \frac{x-y}{2} \\[3pt]& \sin x-\sin y=2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \\[3pt]& \cos x+\cos y=2 \cos \frac{x+y}{2} \cos \frac{x-y}{2} \\[3pt]& \cos x-\cos y=-2 \sin \frac{x+y}{2} \sin \frac{x-y}{2} \end{aligned} \]
5. 其他公式
\[ \dfrac{\cos \alpha}{1-\sin \alpha}=\dfrac{1+\sin \alpha}{\cos \alpha} \\[6pt]\sin \alpha=\dfrac{\sin 2 \alpha}{2 \cos \alpha} \\[6pt]\cos \alpha=\dfrac{\sin 2 \alpha}{2 \sin \alpha} \\[6pt]\tan \alpha+\tan \beta=\tan (\alpha+\beta)(1-\tan \alpha \tan \beta) \]
三、角的变换常用方法
\[ \alpha=(\alpha+\beta)-\beta \\[6pt]\alpha=\beta-(\beta-\alpha) \\[6pt]\alpha=\frac{1}{2}[(\alpha+\beta)+(\alpha-\beta)] \\[6pt]\alpha=\frac{1}{2}[(\alpha+\beta)-(\beta-\alpha)] \\[6pt]\frac{\alpha+\beta}{2}=\left(\alpha-\frac{\beta}{2}\right)-\left(\frac{\alpha}{2}-\beta\right) \\[6pt]\alpha-\gamma=(\alpha-\beta)+(\beta-\gamma) \]
