三角函数常用公式

平方关系

$$sin^2\alpha+cos^2\alpha=1 \qquad 1+tan^2\alpha=sec^2\alpha \qquad 1+cot^2\alpha=csc^2\alpha$$

倒数关系

$$
sin\alpha\ csc\alpha = 1 \qquad cos\alpha \ sec\alpha = 1 \qquad \tan\alpha \ cot\alpha = 1
$$

商数关系

$$
tan\alpha = {sin\alpha \over cos\alpha} \qquad cot\alpha = {cos\alpha \over sin\alpha}
$$

两角和差公式

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

积化和差公式

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

和差化积公式

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

常用倍角公式

$$
sin2\alpha = 2sin\alpha cos\alpha \qquad
cos2\alpha = cos^2\alpha - sin^2\alpha = 2cos^2\alpha-1=1-2sin^2\alpha \qquad
sin3\alpha = 3sin\alpha-4sin^3\alpha \qquad
cos3\alpha = 4cos^3\alpha-3cos\alpha
$$

降幂公式

$$
sin^2\alpha = {1-cos2\alpha \over 2} \qquad
cos^2\alpha = {1+cos\alpha \over 2} \qquad
sin^3\alpha = {3sin\alpha-sin3\alpha \over 4} \qquad
cos^3\alpha = {3cos\alpha+cos3\alpha \over 4}
$$