三角函数

定义式

$$\text{ 余切：}cotA=\frac{1}{tanA}$$
$$\text{ 正切：}secA=\frac{1}{cosA}$$
$$\text{ 余割：}cscA=\frac{1}{sinA}$$

$$\text{ 反正切：}arctan(tanX)=tan(arctanX)=X$$

诱导公式

• sin ⁡ ( − α ) = − sin ⁡ α
• cos ⁡ ( − α ) = cos ⁡ α
• sin ⁡ ( π 2 − α ) = cos ⁡ α
• cos ⁡ ( π 2 − α ) = sin ⁡ α
• sin ⁡ ( π 2 + α ) = cos ⁡ α
• cos ⁡ ( π 2 + α ) = − sin ⁡ α
• sin ⁡ ( π − α ) = sin ⁡ α
• cos ⁡ ( π − α ) = − cos ⁡ α
• sin ⁡ ( π + α ) = − sin ⁡ α
• cos ⁡ ( π + α ) = − cos ⁡ α

平方关系

$$1+ta{n}^2\alpha =se{c}^2\alpha$$
$$1+co{t}^2\alpha =cs{c}^2\alpha$$
$$si{n}^2\alpha +co{s}^2\alpha =1$$

两角和与差的三角函数

$$sin(\alpha +\beta )=sin\alpha cos\beta +cos\alpha sin\beta$$
$$cos(\alpha +\beta )=cos\alpha cos\beta -sin\alpha sin\beta$$
$$sin(\alpha -\beta )=sin\alpha cos\beta -cos\alpha sin\beta$$
$$cos(\alpha -\beta )=cos\alpha cos\beta +sin\alpha sin\beta$$
$$tan(\alpha +\beta )=\frac{tan\alpha +tan\beta }{1-tan\alpha tan\beta }$$
$$tan(\alpha -\beta )=\frac{tan\alpha -tan\beta }{1+tan\alpha tan\beta }$$

积化和差公式

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

和差化积公式

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

倍角公式

$$sin2\alpha =2sin\alpha cos\alpha$$
$$cos2\alpha =cos{}^2\alpha -sin{}^2\alpha =1-2sin{}^2\alpha =2cos{}^2\alpha -1$$
$$sin3\alpha =-4sin{}^3\alpha +3sin\alpha$$
$$cos3\alpha =4cos{}^3\alpha -3cos\alpha$$
$$sin{}^2\alpha =\frac{1-cos2\alpha }{2}$$
$$cos{}^2\alpha =\frac{1+cos2\alpha }{2}$$
$$tan2\alpha =\frac{2tan\alpha }{1-tan{}^2\alpha }$$
$$cot2\alpha =\frac{cot{}^2\alpha -1}{2cot\alpha }$$

半角公式

$$sin{}^2\frac{\alpha }{2}=\frac{1-cos\alpha }{2}$$
$$cos{}^2\frac{\alpha }{2}=\frac{1+cos\alpha }{2}$$
$$sin\frac{\alpha }{2}=\pm \sqrt{\frac{1-cos\alpha }{2}}$$
$$cos\frac{\alpha }{2}=\pm \sqrt{\frac{1+cos\alpha }{2}}$$
$$tan\frac{\alpha }{2}=\frac{1-cos\alpha }{sin\alpha }=\frac{sin\alpha }{1+cos\alpha }=\pm \sqrt{\frac{1-cos\alpha }{1+cos\alpha }}$$
$$cot\frac{\alpha }{2}=\frac{sin\alpha }{1-cos\alpha }=\frac{1+cos\alpha }{sin\alpha }=\pm \sqrt{\frac{1+cos\alpha }{1-cos\alpha }}$$

万能公式

$$sin\alpha =\frac{2tan\frac{\alpha }{2}}{1+ta{n}^2\frac{\alpha }{2}}$$
$$cos\alpha =\frac{1-ta{n}^2\frac{\alpha }{2}}{1+ta{n}^2\frac{\alpha }{2}}$$

其他公式

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

反三角函数恒等式

$$arcsinx+arccosx=\frac{\pi }{2}$$
$$arctanx+arccotx=\frac{\pi }{2}$$
$$sin(arccosx)=\sqrt{1-x^2}$$
$$cos(arcsinx)=\sqrt{1-x^2}$$
$$sin(arcsinx)=x$$
$$arcsin(sinx)=x$$
$$cos(arccosx)=x$$
$$arccos(cosx)=x$$
$$arccos(-x)=\pi -arccosx$$

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三角函数

定义式