三角函数公式

三角函数

（1）基本关系                   （5）和差公式

\(\sin x - \csc x\)   \(\cos x - \sec x\)   \(\tan x - \cot x\)      \(\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta\)

\(\sin^2 x + \cos^2 x = 1\)                  \(\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\)

\(1 + \tan^2 x = \sec^2 x\)   \(1 + \cot^2 x = \csc^2 x\)       \(\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}\) \(\cot(\alpha \pm \beta) = \frac{\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha}\)

（2）诱导公式                   （6）积化和差公式

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

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

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

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

（3）倍角公式                   （7）和差化积公式

\(\sin 2\alpha = 2\sin \alpha \cos \alpha\)   \(= 1 - 2 \sin^2 \alpha\)        \(\sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}\)

\(\cos 2\alpha = \cos^2 \alpha-\sin^2 \alpha=2 \cos^2 \alpha - 1\)       \(\sin \alpha - \sin \beta = 2 \sin \frac{\alpha - \beta}{2} \cos \frac{\alpha + \beta}{2}\)

\(\sin 3\alpha = -4 \sin^3 \alpha + 3 \sin \alpha\)             \(\cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}\)

\(\cos 3 \alpha = 4 \cos^3 \alpha - 3 \cos \alpha\)             \(\cos \alpha - \cos \beta = - 2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2}\)

\(\tan 2\alpha = \frac{2\tan \alpha}{1 - \tan^2 \alpha}\)      \(\cot 2\alpha = \frac{\cot^2 \alpha - 1}{2 \cot \alpha}\)

（4）半角公式                   （8）万能公式

\(\sin^2 \frac{\alpha}{2} = \frac{1}{2}(1 - \cos \alpha)\)  \(\cos^2 \frac{\alpha}{2} = \frac{1}{2}(1 + \cos \alpha)\)   \(u = \tan \frac{x}{2}\),\(\sin x = \frac{2u}{1 + u^2}\), \(\cos x = \frac{1 - u^2}{1 + u^2}\)

\(\sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}}\)     \(\cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}}\)

\(\tan \frac{\alpha}{2} = \frac{1 - \cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1 + \cos \alpha} = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}}\)

\(\cot \frac{\alpha}{2} = \frac{\sin \alpha}{1 - \cos \alpha} = \frac{1 + \cos \alpha}{\sin \alpha} = \pm \sqrt{\frac{1 + \cos \alpha}{1 - \cos \alpha}}\)