三角函数恒等变换

三角函数

基本操作

\(\sin^2\alpha+\cos^2\alpha=1\)
\(\tan\alpha=\Large\frac{\sin\alpha}{\cos\alpha}\)

诱导公式

角	\(\alpha+2k\pi\)	\(\pi+\alpha\)	\(-\alpha\)	\(\pi-\alpha\)	\(\frac{\pi}{2}-\alpha\)	\(\frac{\pi}{2}+\alpha\)
正弦	\(\sin\alpha\)	\(-\sin\alpha\)	\(\sin\alpha\)	\(\sin\alpha\)	\(\cos\alpha\)	\(\cos\alpha\)
余弦	\(\cos\alpha\)	\(-\cos\alpha\)	\(\cos\alpha\)	\(-\cos\alpha\)	\(\sin\alpha\)	\(-\sin\alpha\)
正切	\(\tan\alpha\)	\(\tan\alpha\)	\(-\tan\alpha\)	\(-\tan\alpha\)	\(\frac{1}{\tan\alpha}\)	\(-\frac{1}{\tan\alpha}\)

两角和，差公式

\(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\)
\(\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\)
\(\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\)
\(\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\)

二倍角

\(\sin2\alpha=2\sin\alpha\cos\alpha\)
\( \cos2\alpha= \left\{\begin{matrix} \cos^2\alpha-\sin^2\alpha\\ 1-2\sin^2\alpha\\ 2\cos^2\alpha-1 \end{matrix}\right. \)
\(\tan2\alpha=\Large {2\tan\alpha\over1-tan^2\alpha}\)