[安乐椅#18] 三角函数公式（不）大全

特殊角三角函数值

$\sin{\dfrac{\pi}{12}}=\dfrac{\sqrt{6}-\sqrt{2}}{4} \space\space\space\space\space\space\space\space \cos{\dfrac{\pi}{12}}=\dfrac{\sqrt{6}+\sqrt{2}}{4} \space\space\space\space\space\space\space\space \tan{\dfrac{\pi}{12}}=2-\sqrt{3}$

$\sin{\dfrac{\pi}{8}}=\dfrac{\sqrt{2-\sqrt{2}}}{2} \space\space\space\space\space\space\space\space\space \cos{\dfrac{\pi}{8}}=\dfrac{\sqrt{2+\sqrt{2}}}{2} \space\space\space\space\space\space\space\space\space \tan{\dfrac{\pi}{8}}=\sqrt{2}-1$

$\sin{\dfrac{\pi}{6}}=\dfrac{1}{2} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \cos{\dfrac{\pi}{6}}=\dfrac{\sqrt{3}}{2} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \tan{\dfrac{\pi}{6}}=\dfrac{\sqrt{3}}{3}$

$\sin{\dfrac{\pi}{4}}=\dfrac{\sqrt{2}}{2} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \cos{\dfrac{\pi}{4}}=\dfrac{\sqrt{2}}{2} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \tan{\dfrac{\pi}{4}}=1$

$\sin{\dfrac{\pi}{3}}=\dfrac{\sqrt{3}}{2} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \cos{\dfrac{\pi}{3}}=\dfrac{1}{2} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \tan{\dfrac{\pi}{3}}=\sqrt{3}$

$\sin{\dfrac{3\pi}{8}}=\dfrac{\sqrt{2+\sqrt{2}}}{2} \space\space\space\space\space\space\space \cos{\dfrac{3\pi}{8}}=\dfrac{\sqrt{2-\sqrt{2}}}{2} \space\space\space\space\space\space\space \tan{\dfrac{3\pi}{8}}=\sqrt{2}+1$

$\sin{\dfrac{5\pi}{12}}=\dfrac{\sqrt{6}+\sqrt{2}}{4} \space\space\space\space\space\space\space\space \cos{\dfrac{5\pi}{12}}=\dfrac{\sqrt{6}-\sqrt{2}}{4} \space\space\space\space\space\space\space \tan{\dfrac{5\pi}{12}}=2+\sqrt{3}$

$\sin{\dfrac{\pi}{2}}=1 \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \cos{\dfrac{\pi}{2}}=0$

辅助角公式

$$a\sin x+b \cos x=\sqrt{a^2+b^2}\sin(x+ \varphi)$$

$$\text{其中，}\cos{\varphi}=\dfrac{a}{\sqrt{a^2+b^2}},\space \sin{\varphi}=\dfrac{b}{\sqrt{a^2+b^2}}$$

和差化积

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

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

$$\cos\alpha+\cos\beta=2\cos\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha-\beta}{2}$$

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

$$\tan\alpha+\tan\beta=\dfrac{\sin{(\alpha+\beta)}}{\cos{\alpha}\cos{\beta}}$$

$$\tan\alpha-\tan\beta=\dfrac{\sin{(\alpha-\beta)}}{\cos{\alpha}\cos{\beta}}$$

$$\cot\alpha+\cot\beta=\dfrac{\sin{(\alpha+\beta)}}{\sin{\alpha}\sin{\beta}}$$

$$\cot\alpha-\cot\beta=-\dfrac{\sin{(\alpha-\beta)}}{\sin{\alpha}\sin{\beta}}$$

积化和差

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

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

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

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