物理

三角函数公式合集

${（以下均默认式子有意义）}$

${1.几个基本的结论}$

${1.1~\sin^2\theta+\cos^2\theta=1,\frac{\sin\theta}{\cos\theta}=\tan\theta}$

${1.2~\tan^2\theta+1=\frac{1}{\cos^2\theta},\frac{1}{\tan^2\theta}+1=\frac{1}{\sin^2\theta}}$

${2.诱导公式}$

${2.1~\sin(\theta+2k\pi)=\sin\theta,\cos(\theta+2k\pi)=\cos \theta, \tan(\theta+2k\pi)=\tan \theta (k \in\mathbb{Z})}$

${2.2~\sin(\theta+\pi)=-\sin \theta,\cos(\theta+\pi)=-\cos \theta, \tan(\theta+\pi)=\tan \theta}$

${2.3~\sin(-\theta)=-\sin \theta,\cos(-\theta)=\cos \theta, \tan(-\theta)=-\tan \theta}$

${2.4~\sin(\pi-\theta)=\sin \theta,\cos(\pi-\theta)=-\cos \theta, \tan(\pi-\theta)=-\tan \theta}$

${2.5~\sin(\theta+\frac{\pi}{2})=\cos\theta, \cos (\theta+\frac{\pi}{2})=-\sin \theta,\tan(\theta+\frac{\pi}{2})=-\frac{1}{\tan \theta}}$

${2.6~\sin(\theta-\frac{\pi}{2})=-\cos\theta, \cos (\theta-\frac{\pi}{2})=\sin \theta,\tan(\theta-\frac{\pi}{2})=-\frac{1}{\tan \theta}}$

${3.和差角公式}$

${\sin(\alpha \pm \beta)=\sin \alpha \cos\beta \pm \cos \alpha\sin\beta}$

${\cos(\alpha \pm \beta)=\cos \alpha \cos\beta \mp \sin \alpha \sin \beta}$

${\tan(\alpha \pm \beta)=\frac{\tan \alpha\pm \tan \beta}{1 \mp \tan \alpha \tan \beta}}$

${4.倍角公式}$

${4.1~\sin 2\theta=2\sin \theta \cos \theta}$

${ \cos2\theta=2\cos^2\theta-1=1-2\sin^2\theta=\cos^2\theta-\sin^2\theta}$

${\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}}$

${4.2~\sin 3\theta=3\sin\theta-4\sin^3\theta=4\sin\theta\sin(\theta+\frac{\pi}{3})\sin(\frac{\pi}{3}-\theta)}$

${\cos 3\theta=3\cos\theta+4\cos^3\theta=4\cos \theta \cos (\theta+\frac{\pi}{3})\cos(\frac{\pi}{3}-\theta)}$

${\tan 3\theta=\tan\theta\tan(\theta+\frac{\pi}{3})\tan(\frac{\pi}{3}-\theta)}$

${5.半角公式}$

${5.1~\sin^2\theta=\frac{1-\cos2\theta}{2}, \cos^2 \theta=\frac{1+\cos 2\theta}{2}, \tan^2\theta=\frac{1-\cos2\theta}{1+\cos 2\theta}}$

${5.2~\tan \theta=\frac{\sin 2\theta}{\cos2\theta+1}=\frac{1-\cos 2\theta}{\sin 2\theta}}$

${6.辅助角公式}$

${A\sin\theta+B\cos\theta=\sqrt{A^2+B^2}\sin(\theta+\phi),其中\sin \phi=\frac{B}{\sqrt{A^2+B^2}}, \cos\phi=\frac{A}{\sqrt{A^2+B^2}}}$

${7.和差化积公式}$

${\sin \alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}}$

${\sin \alpha-\sin\beta=2\sin\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}}$

${\cos \alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}}$

${\cos \alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}}$

${8.积化和差公式}$

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

${\cos \alpha \cos\beta=\frac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)]}$

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

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

${9.万能公式}$

${\sin 2\theta=\frac{2\tan \theta}{1+\tan^2\theta}, \cos 2\theta=\frac{1-\tan^2\theta}{1+\tan^2\theta},\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}}$