5.5.1 两角和与差的正弦、余弦和正切公式

5.5 三角恒等变换

知识要点

• $C_{(\alpha-\beta)}$

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

• $C_{(\alpha+\beta)}$、$S_{(\alpha+\beta)}$、$S_{(\alpha-\beta)}$

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

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

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

• $T_{(\alpha+\beta)}$、$T_{(\alpha-\beta)}$

$\tan(\alpha+\beta)=\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$

$\tan(\alpha-\beta)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$

• $S_{2\alpha}$、$C_{2\alpha}$、$T_{2\alpha}$

$\sin2\alpha=2\sin\alpha\cos\alpha$

$\cos2\alpha=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1$

$\tan2\alpha=\dfrac{2\tan\alpha}{1-\tan^2\alpha}$