[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

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

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

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

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