三角函数恒等变形
基本性质
\[\sin^2\alpha+\cos^2\alpha=1
\]
\[\tan\alpha=\frac{\sin\alpha}{\cos\alpha}
\]
和差角公式
\[\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta
\]
\[\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta
\]
\[\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}
\]
二倍角公式
\[\sin2\alpha=2\sin\alpha\cos\alpha
\]
\[\cos2\alpha=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1
\]
\[\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}
\]
降幂公式
\[\sin\alpha\cos\alpha=\frac12\sin2\alpha
\]
\[\sin^2\alpha=\frac{1-\cos2\alpha}2
\]
\[\cos^2\alpha=\frac{1+\cos2\alpha}2
\]
一点也不万能的公式
\[\sin\alpha=\frac{2\tan\frac\alpha2}{1+\tan^2\frac\alpha2}
\]
\[\cos\alpha=\frac{1-\tan^2\frac\alpha2}{1+\tan^2\frac\alpha2}
\]
积化和差
\[\sin\alpha\cos\beta=\frac12\left[\sin(\alpha+\beta)+\sin(\alpha-\beta)\right]
\]
\[\cos\alpha\sin\beta=\frac12\left[\sin(\alpha+\beta)-\sin(\alpha-\beta)\right]
\]
\[\sin\alpha\sin\beta=-\frac12\left[\cos(\alpha+\beta)-\cos(\alpha-\beta)\right]
\]
\[\cos\alpha\cos\beta=\frac12\left[\cos(\alpha+\beta)+\cos(\alpha-\beta)\right]
\]
\[\tan\alpha\tan\beta=1-\frac{\tan\alpha+\tan\beta}{\tan(\alpha+\beta)}=\frac{\tan\alpha-\tan\beta}{\tan(\alpha-\beta)}-1
\]
和差化积
\[\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2
\]
\[\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2
\]
\[\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2
\]
\[\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2
\]
\[\tan\alpha\pm\tan\beta=\tan(\alpha\pm\beta)(1\mp\tan\alpha\tan \beta)
\]
杂题
\[\begin{align*}
&\cos75°\\
={}&\cos(30°+45°)\\
={}&\cos30°\cos45°-\sin30°\sin45°\\
={}&\frac{\sqrt3}2\times\frac{\sqrt2}2-\frac12\times\frac{\sqrt2}2\\
={}&\frac{\sqrt6-\sqrt2}2
\end{align*}
\]
\[\begin{align*}
&\sin105°\\
={}&\sin(45°+60°)\\
={}&\sin45°\cos60°+\cos45°\sin60°\\
={}&\frac{\sqrt2}2\times\frac12+\frac{\sqrt2}2\times\frac{\sqrt3}2\\
={}&\frac{\sqrt6+\sqrt2}2
\end{align*}
\]
\[\begin{align*}
&\cos61°\cos16°+\sin61°\sin16°\\
={}&\cos(61°-16°)\\
={}&\cos45°\\
={}&\frac{\sqrt2}2
\end{align*}
\]
\[\begin{align*}
&\sin13°\cos17°+\cos13°\sin17°\\
={}&\sin(13°+17°)\\
={}&\sin30°\\
={}&\frac12
\end{align*}
\]
\[\begin{align*}
&\sin163°\sin223°+\sin253°\sin313°\\
={}&\sin163°\sin223°+\sin(163°+90°)\sin(223°+90°)\\
={}&\sin163°\sin223°+\cos163°\cos223°\\
={}&\cos(223°-163°)\\
={}&\cos(-60°)\\
={}&\cos60°\\
={}&\frac12
\end{align*}
\]
\[\begin{align*}
&\cos43°\cos77°+\sin43°\cos167°\\
={}&\cos43°\cos77°+\sin43°\cos(77°+90°)\\
={}&\cos43°\cos77°-\sin43°\sin77°\\
={}&\cos(43°+77°)\\
={}&\cos120°\\
={}&-\sin30°\\
={}&-\frac12\\
\end{align*}
\]
\[\begin{align*}
&\frac{\sin7°+\cos15°\sin8°}{\cos7°-\sin15°\sin8°}\\
={}&\frac{\sin(15°-8°)+\cos15°\sin8°}{\cos(15°-8°)-\sin15°\sin8°}\\
={}&\frac{\sin15°\cos8°-\cos15°\sin8°+\cos15°\sin8°}{\cos15°\cos8°+\sin15°\sin8°-\sin15°\sin8°}\\
={}&\frac{\sin15°\cos8°}{\cos15°\cos8°}\\
={}&\tan15°\\
={}&\tan(45°-30°)\\
={}&\frac{\tan45°-\tan30°}{1+\tan45°\tan30°}\\
={}&\frac{1-\frac{\sqrt3}3}{1+1\times\frac{\sqrt3}3}\\
={}&2-\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\frac{1+\tan75°}{1-\tan75°}\\
={}&\frac{\tan45°+\tan75°}{1-\tan45°\tan75°}\\
={}&\tan(45°+75°)\\
={}&\tan120°\\
={}&-\tan60°\\
={}&-\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\frac{\cos15°-\sin15°}{\cos15°+\sin15°}\\
={}&\frac{1-\tan15°}{1+\sin15°}\\
={}&\frac{\tan45°-\tan15°}{1+\tan45°\tan15°}\\
={}&\tan30°\\
={}&\frac{\sqrt3}3
\end{align*}
\]
\[\begin{align*}
&\sin15°\cos15°\\
={}&\frac12\sin30°\\
={}&\frac12\times\frac12\\
={}&\frac14\\
\end{align*}
\]
\[\begin{align*}
&1-2\sin^275°\\
={}&\cos(2\times75°)\\
={}&\cos150°\\
={}&-\cos60°\\
={}&-\frac12\\
\end{align*}
\]
\[\begin{align*}
&1-2\sin^275°\\
={}&\cos(2\times75°)\\
={}&\cos150°\\
={}&-\cos60°\\
={}&-\frac12\\
\end{align*}
\]
\[\begin{align*}
&1-2\sin^275°\\
={}&\cos(2\times75°)\\
={}&\cos150°\\
={}&-\cos60°\\
={}&-\frac12\\
\end{align*}
\]
\[\begin{align*}
&\frac{2\tan150°}{1-\tan^2150°}\\
={}&\tan(2\times150°)\\
={}&\tan300°\\
={}&-\tan60°\\
={}&-\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\tan15°+\cot15°\\
={}&\frac{\sin15°}{\cos15°}+\frac{\cos15°}{\sin15°}\\
={}&\frac{\sin^215°+\cos^215°}{\sin15°\cos15°}\\
={}&\frac1{\sin15°\cos15°}\\
={}&\frac1{\frac12\sin30°}\\
={}&\frac1{\frac12\times\frac12}\\
={}&4\\
\end{align*}
\]
\[\begin{align*}
&\frac{2\cos10°-\sin20°}{\sin70°}\\
={}&\frac{2\cos(30°-20°)-\sin20°}{\cos20°}\\
={}&\frac{2(\cos30°\cos20°+\sin30°\sin20°)-\sin20°}{\cos20°}\\
={}&\frac{2(\frac{\sqrt3}2\times\cos20°+\frac12\times\sin20°)-\sin20°}{\cos20°}\\
={}&\frac{\sqrt3\cos20°+\sin20°-\sin20°}{\cos20°}\\
={}&\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\tan20°+4\sin20°\\
={}&\frac{\sin20°}{\cos20°}+\frac{4\sin20°\cos20°}{\cos20°}\\
={}&\frac{\sin20°+2\sin40°}{\cos20°}\\
={}&\frac{\sin20°+2\sin(60°-20°)}{\cos20°}\\
={}&\frac{\sin20°+2\sin60°\cos20°-2\cos60°\sin20°}{\cos20°}\\
={}&\frac{\sin20°+\sqrt3\cos20°-\sin20°}{\cos20°}\\
={}&\sqrt3
\end{align*}\\
\]
\[\begin{align*}
&\tan72°-\tan42°-\frac{\sqrt3}3\tan72°\tan42°\\
={}&\tan(72°-42°)(1+\tan72°\tan42°)-\tan30°\tan72°\tan42°\\
={}&\tan30°(1+\tan72°\tan42°)-\tan30°(\tan72°\tan42°)\\
={}&\tan30°\\
={}&\frac{\sqrt3}3\\
\end{align*}
\]
\[\begin{align*}
&\frac{\tan20°+\tan40°+\tan120°}{\tan20°\tan40°}\\
={}&\frac{\tan60°-\tan20°\tan40°\tan60°-\tan60°}{\tan20°\tan40°}\\
={}&-\tan60°\\
={}&-\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\tan(30°-\alpha)+\tan(30°+\alpha)+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\
={}&\tan(30°-\alpha+30°+\alpha)\left[1-\tan(30°+\alpha)\tan(30°-\alpha)\right]+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\
={}&\sqrt3\left[1-\tan(30°+\alpha)\tan(30°-\alpha)\right]+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\
={}&\sqrt3-\sqrt3\tan(30°+\alpha)\tan(30°-\alpha)+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\
={}&\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\tan10°(3+\tan30°\tan40°+\tan40°\tan50°+\tan50°\tan60°)\\
={}&\tan10°\left[(1+\tan30°\tan40°)+(1+\tan40°\tan50°)+(1+\tan50°\tan60°)\right]\\
={}&\tan10°\left[(1+\frac{\tan40°-\tan30°}{\tan10°}-1)+(1+\frac{\tan50°-\tan40°}{\tan10°}-1)+(1+\frac{\tan60°-\tan50°}{\tan10°}-1)\right]\\
={}&\tan10°(\frac{\tan40°-\tan30°}{\tan10°}+\frac{\tan50°-\tan40°}{\tan10°}+\frac{\tan60°-\tan50°}{\tan10°})\\
={}&\tan40°-\tan30°+\tan50°-\tan40°+\tan60°-\tan50°\\
={}&\tan60°-\tan30°\\
={}&\tan60°-\tan30°\\
={}&\frac{2\sqrt3}3\\
\end{align*}
\]
\[\begin{align*}
&\frac{\sin50°(1+\sqrt3\tan10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{\sin50°(\frac{\cos10°}{\cos10°}+\frac{\sqrt3\sin10°}{\cos10°})-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{\frac{\sin50°}{\cos10°}\times(\sqrt3\sin10°+\cos10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{\frac{\sin50°}{\cos10°}\times2(\frac{\sqrt3}2\sin10°+\frac12\cos10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{\frac{\sin50°}{\cos10°}\times2(\cos30°\sin10°+\sin30°\cos10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{\frac{\sin50°}{\cos10°}\times2\sin(30°+10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{\frac{\cos40°}{\sin80°}\times2\sin40°-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{1-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\
={}&\frac{1-(1-2\sin^210°)}{\cos80°\sqrt{1-(1-2\sin^210°)}}\\
={}&\frac{2\sin^210°}{\cos80°\sqrt{2\sin^210°}}\\
={}&\frac{2\sin^210°}{\sqrt2\sin^210°}\\
={}&\sqrt2\\
\end{align*}
\]
\[\begin{align*}
&(\tan10°-\sqrt3)\times\frac{\cos10°}{\sin50°}\\
={}&(\tan10°-\tan60°)\times\frac{\cos10°}{\sin50°}\\
={}&\left[\tan(10°-60°)\right](1+\tan10°\tan60°)\times\frac{\cos10°}{\sin50°}\\
={}&-\tan50°(1+\tan10°\tan60°)\times\frac{\cos10°}{\sin50°}\\
={}&-\tan50°(1+\sqrt3\tan10°)\times\frac{\cos10°}{\sin50°}\\
={}&-2\tan50°(\frac12\times\frac{\cos10°}{\cos10°}+\frac{\sqrt3}2\times\frac{\sin10}{\cos10})\times\frac{\cos10°}{\sin50°}\\
={}&-2\tan50°(\cos60°\cos10°+\sin60°\sin10)\times\frac1{\sin50°}\\
={}&-2\tan50°(\cos60°-10°)\times\frac1{\sin50°}\\
={}&-2\tan50°(\cos60°-10°)\times\frac1{\sin50°}\\
={}&-2
\end{align*}
\]
\[\begin{align*}
&\sqrt{\sin^280°}\left[2\sin50°+\sin10°(1+\sqrt3\tan10°)\right]\\
={}&\sqrt{\sin^280°}\left[2\sin50°+2\sin10°(\frac12+\frac{\sqrt3}2\tan10°)\right]\\
={}&2\sqrt{\sin^280°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°\cos10°+\sin60°\sin10°)\right]\\
={}&2\sqrt{\sin^280°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°-10°)\right]\\
={}&2\cos10°(\sin50°+\tan10°\cos50°)\\
={}&2(\sin50°\cos10°+\cos50°\sin10°)\\
={}&2\sin(50°+10°)\\
={}&\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\sqrt{1+\cos20°}\left[2\sin50°+\sin10°(1+\sqrt3\tan10°)\right]\\
={}&\sqrt{1+\cos20°}\left[2\sin50°+2\sin10°(\frac12+\frac{\sqrt3}2\tan10°)\right]\\
={}&2\sqrt{1+\cos20°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°\cos10°+\sin60°\sin10°)\right]\\
={}&2\sqrt{1+\cos20°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°-10°)\right]\\
={}&2\sqrt{1+\cos20°}(\sin50°+\tan10°\cos50°)\\
={}&2\sqrt{1+2\cos^210°-1}(\sin50°+\tan10°\cos50°)\\
={}&2\sqrt{1+2\cos^210°-1}(\sin50°+\tan10°\cos50°)\\
={}&2\sqrt2\cos10°(\sin50°+\tan10°\cos50°)\\
={}&2\sqrt2\sin60°\\
={}&\sqrt6\\
\end{align*}
\]
\[\begin{align*}
&4\cos50°-\tan40°\\
={}&4\sin40°-\frac{\sin40°}{\cos40°}\\
={}&\frac{4\sin40°\cos40°-\sin40°}{\cos40°}\\
={}&\frac{2\sin80°-\sin40°}{\cos40°}\\
={}&\frac{2\cos10°-\sin(10°+30°)}{\cos40°}\\
={}&\frac{2\cos10°-(\sin10°\cos30°+\cos10°\sin30°)}{\cos40°}\\
={}&\frac{2\cos10°-(\frac{\sqrt3}2\sin10°+\frac12\cos10°)}{\cos40°}\\
={}&\frac{\frac32\cos10°-\frac{\sqrt3}2\sin10°}{\cos40°}\\
={}&\sqrt3\times\frac{\frac{\sqrt3}2\cos10°-\frac12\sin10°}{\cos40°}\\
={}&\sqrt3\times\frac{\cos30°\cos10°-\sin30°\sin10°}{\cos40°}\\
={}&\sqrt3\times\frac{\cos(30°+10°)}{\cos40°}\\
={}&\sqrt3\\
\end{align*}\\
\]
\[\begin{align*}
&\frac{\sqrt3\tan12°-3}{4\cos^212°\sin12°-2\sin12°}\\
={}&\sqrt3\times\frac{\tan12°-\tan60°}{4\cos^212°\sin12°-2\sin12°}\\
={}&\sqrt3\times\frac{\frac{\sin12°\cos60°-\cos12°\sin60°}{\cos12°\cos60°}}{4\cos^212°\sin12°-2\sin12°}\\
={}&\sqrt3\times\frac{\frac{\sin(12°-60°)}{\frac12\cos12°}}{4\cos^212°\sin12°-2\sin12°}\\
={}&\sqrt3\times\frac{\frac{-2\sin48°}{\cos12°}}{4\cos^212°\sin12°-2\sin12°}\\
={}&\sqrt3\times\frac{-2\sin48°}{2\sin12°\cos12°(2\cos^212°-1)}\\
={}&\sqrt3\times\frac{-2\sin48°}{\sin24°\cos24°}\\
={}&\sqrt3\times\frac{-2\sin48°}{\frac12\sin48°}\\
={}&-4\sqrt3\\
\end{align*}
\]
\[\begin{align*}
&\frac1{\sin10°}-\frac{\sqrt3}{\cos10°}\\
={}&\frac{\cos10°-\sqrt3\sin10°}{\sin10°\cos10°}\\
={}&2\times\frac{\frac12\cos10°-\frac{\sqrt3}2\sin10°}{\sin10°\cos10°}\\
={}&2\times\frac{\sin30°\cos10°-\cos30°\sin10°}{\sin10°\cos10°}\\
={}&2\times\frac{\sin(30°-10°)}{\frac12\sin20°}\\
={}&4\\
\end{align*}
\]
\[\begin{align*}
&\cos^4x-2\sin x\cos x-\sin^4x\\
={}&(\cos^2x+\sin^2x)(\cos^2x-\sin^2x)-2\sin x\cos x\\
={}&\cos2x-\sin2x\\
={}&\sqrt2(\frac{\sqrt2}2\cos2x-\frac{\sqrt2}2\sin2x)\\
={}&\sqrt2(\sin\frac\pi4\cos2x-\cos\frac\pi4\sin2x)\\
={}&\sqrt2(\sin\frac\pi4\cos2x-\cos\frac\pi4\sin2x)\\
={}&\sqrt2\sin(\frac\pi4-2x)\\
={}&\sqrt2\sin(2x-\frac{3\pi}4)\\
\end{align*}
\]
\[\begin{align*}
&\sin(x+\frac\pi6)+\sin(x-\frac\pi6)+\cos x\\
={}&\sin(x+\frac\pi6)+\sin x\cos\frac\pi6-\cos x\sin\frac\pi6+\cos x\\
={}&\sin(x+\frac\pi6)+\sin x\cos\frac\pi6+\cos x\sin\frac\pi6\\
={}&\sin(x+\frac\pi6)+\sin(x+\frac\pi6)\\
={}&2\sin(x+\frac\pi6)\\
\end{align*}
\]
\[\begin{align*}
&a\sin x+b\cos x\\
={}&\sqrt{a^2+b^2}(\frac{a}{\sqrt{a^2+b^2}}\sin x+\frac{b}{\sqrt{a^2+b^2}}\cos x)\\
={}&\sqrt{a^2+b^2}(\sin x\cos\alpha+\cos x\sin\alpha)\\
={}&\sqrt{a^2+b^2}\sin(x+\alpha)
\end{align*}
\]
