数学整理1

三角函数的基本公式

• 基本恒等式
• 奇偶性
• 周期性
• 二倍角
• 三倍角
• 和差公式
• 半角公式
• 和差化积
• 积化和差

$$\sin^2\theta + \cos^2\theta = 1$$

$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

$$\cot\theta = \frac{\cos\theta}{\sin\theta}$$

$$\sec\theta = \frac{1}{\cos\theta}$$

$$\csc\theta = \frac{1}{\sin\theta}$$

$$\sin(-\theta) = -\sin\theta$$

$$\cos(-\theta) = \cos\theta$$

$$\tan(-\theta) = -\tan\theta$$

$$\cot(-\theta) = -\cot\theta$$

$$\sec(-\theta) = \sec\theta$$

$$\csc(-\theta) = -\csc\theta$$

$$\sin(\theta + 2k\pi) = \sin\theta$$

$$\cos(\theta + 2k\pi) = \cos\theta$$

$$\tan(\theta + k\pi) = \tan\theta$$

$$\cot(\theta + k\pi) = \cot\theta$$

$$\sec(\theta + 2k\pi) = \sec\theta$$

$$\csc(\theta + 2k\pi) = \csc\theta$$

其中$k$是任意整数。

$$\sin2\theta=2\sin\theta\cos\theta$$

$$\cos2\theta=\cos^2\theta-\sin^2\theta$$

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

$$\sin(3A) = 3\sin(A) - 4\sin^3(A)$$

$$\cos(3A) = 4\cos^3(A) - 3\cos(A)$$

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

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

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

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

$$\cos^2{\frac{\theta}{2}} = {\frac{1 + \cos{\theta}}{2}}$$

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

$$\sin A + \sin B = {2}\sin{\frac{A+B}{2}}\cos{\frac{A-B}{2}}$$

$$\sin A - \sin B = {2}\cos{\frac{A+B}{2}}\sin{\frac{A-B}{2}}$$

$$\cos A + \cos B = {2}\cos{\frac{A+B}{2}}\cos{\frac{A-B}{2}}$$

$$\cos A - \cos B = {-2}\sin{\frac{A+B}{2}}\sin{\frac{A-B}{2}}$$

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

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

$$\cos A \cos B = {\frac{1}{2}}[\cos(A+B)+ \cos(A-B)]$$

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