标签：cos frac 三角函数 变形 beta sin notag alpha 恒等
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
\[\sin\alpha\cos\beta=\frac12\left[\sin(\alpha+\beta)+\sin(\alpha-\beta)\right]
\]
\[\cos\alpha\sin\beta=\frac12\left[\sin(\alpha+\beta)-\sin(\alpha-\beta)\right]
\]
\[\sin\alpha\sin\beta=-\frac12\left[\cos(\alpha+\beta)-\cos(\alpha-\beta)\right]
\]
\[\cos\alpha\cos\beta=\frac12\left[\cos(\alpha+\beta)+\cos(\alpha-\beta)\right]
\]
和差化积
\[\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2
\]
\[\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2
\]
\[\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2
\]
\[\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2
\]
杂题
\[\begin{align*}
&\cos75°\notag\\
={}&\cos(30°+45°)\notag\\
={}&\cos30°\cos45°-\sin30°\sin45°\notag\\
={}&\frac{\sqrt3}2\times\frac{\sqrt2}2-\frac12\times\frac{\sqrt2}2\notag\\
={}&\frac{\sqrt6-\sqrt2}2
\end{align*}
\]
\[\begin{align*}
&\sin105°\notag\\
={}&\sin(45°+60°)\notag\\
={}&\sin45°\cos60°+\cos45°\sin60°\notag\\
={}&\frac{\sqrt2}2\times\frac12+\frac{\sqrt2}2\times\frac{\sqrt3}2\notag\\
={}&\frac{\sqrt6+\sqrt2}2
\end{align*}
\]
\[\begin{align*}
&\cos61°\cos16°+\sin61°\sin16°\notag\\
={}&\cos(61°-16°)\notag\\
={}&\cos45°\notag\\
={}&\frac{\sqrt2}2
\end{align*}
\]
\[\begin{align*}
&\sin13°\cos17°+\cos13°\sin17°\notag\\
={}&\sin(13°+17°)\notag\\
={}&\sin30°\notag\\
={}&\frac12
\end{align*}
\]
\[\begin{align*}
&\sin163°\sin223°+\sin253°\sin313°\notag\\
={}&\sin163°\sin223°+\sin(163°+90°)\sin(223°+90°)\notag\\
={}&\sin163°\sin223°+\cos163°\cos223°\notag\\
={}&\cos(223°-163°)\notag\\
={}&\cos(-60°)\notag\\
={}&\cos60°\notag\\
={}&\frac12
\end{align*}
\]
\[\begin{align*}
&\cos43°\cos77°+\sin43°\cos167°\notag\\
={}&\cos43°\cos77°+\sin43°\cos(77°+90°)\notag\\
={}&\cos43°\cos77°-\sin43°\sin77°\notag\\
={}&\cos(43°+77°)\notag\\
={}&\cos120°\notag\\
={}&-\sin30°\notag\\
={}&-\frac12\notag\\
\end{align*}
\]
\[\begin{align*}
&\frac{\sin7°+\cos15°\sin8°}{\cos7°-\sin15°\sin8°}\notag\\
={}&\frac{\sin(15°-8°)+\cos15°\sin8°}{\cos(15°-8°)-\sin15°\sin8°}\notag\\
={}&\frac{\sin15°\cos8°-\cos15°\sin8°+\cos15°\sin8°}{\cos15°\cos8°+\sin15°\sin8°-\sin15°\sin8°}\notag\\
={}&\frac{\sin15°\cos8°}{\cos15°\cos8°}\notag\\
={}&\tan15°\notag\\
={}&\tan(45°-30°)\notag\\
={}&\frac{\tan45°-\tan30°}{1+\tan45°\tan30°}\notag\\
={}&\frac{1-\frac{\sqrt3}3}{1+1\times\frac{\sqrt3}3}\notag\\
={}&2-\sqrt3\notag\\
\end{align*}
\]
\[\begin{align*}
&\frac{1+\tan75°}{1-\tan75°}\notag\\
={}&\frac{\tan45°+\tan75°}{1-\tan45°\tan75°}\notag\\
={}&\tan(45°+75°)\notag\\
={}&\tan120°\notag\\
={}&-\tan60°\notag\\
={}&-\sqrt3\notag\\
\end{align*}
\]
\[\begin{align*}
&\frac{\cos15°-\sin15°}{\cos15°+\sin15°}\notag\\
={}&\frac{1-\tan15°}{1+\sin15°}\notag\\
={}&\frac{\tan45°-\tan15°}{1+\tan45°\tan15°}\notag\\
={}&\tan30°\notag\\
={}&\frac{\sqrt3}3
\end{align*}
\]
\[\begin{align*}
&\sin15°\cos15°\notag\\
={}&\frac12\sin30°\notag\\
={}&\frac12\times\frac12\notag\\
={}&\frac14\notag\\
\end{align*}
\]
\[\begin{align*}
&1-2\sin^275°\notag\\
={}&\cos(2\times75°)\notag\\
={}&\cos150°\notag\\
={}&-\cos60°\notag\\
={}&-\frac12\notag\\
\end{align*}
\]
\[\begin{align*}
&1-2\sin^275°\notag\\
={}&\cos(2\times75°)\notag\\
={}&\cos150°\notag\\
={}&-\cos60°\notag\\
={}&-\frac12\notag\\
\end{align*}
\]
\[\begin{align*}
&1-2\sin^275°\notag\\
={}&\cos(2\times75°)\notag\\
={}&\cos150°\notag\\
={}&-\cos60°\notag\\
={}&-\frac12\notag\\
\end{align*}
\]
\[\begin{align*}
&\frac{2\tan150°}{1-\tan^2150°}\notag\\
={}&\tan(2\times150°)\notag\\
={}&\tan300°\notag\\
={}&-\tan60°\notag\\
={}&-\sqrt3\notag\\
\end{align*}
\]
\[\begin{align*}
&\tan15°+\cot15°\notag\\
={}&\frac{\sin15°}{\cos15°}+\frac{\cos15°}{\sin15°}\notag\\
={}&\frac{\sin^215°+\cos^215°}{\sin15°\cos15°}\notag\\
={}&\frac1{\sin15°\cos15°}\notag\\
={}&\frac1{\frac12\sin30°}\notag\\
={}&\frac1{\frac12\times\frac12}\notag\\
={}&4\notag\\
\end{align*}
\]
\[\begin{align*}
&\frac{2\cos10°-\sin20°}{\sin70°}\notag\\
={}&\frac{2\cos(30°-20°)-\sin20°}{\cos20°}\notag\\
={}&\frac{2(\cos30°\cos20°+\sin30°\sin20°)-\sin20°}{\cos20°}\notag\\
={}&\frac{2(\frac{\sqrt3}2\times\cos20°+\frac12\times\sin20°)-\sin20°}{\sin20°}\notag\\
={}&\frac{\sqrt3\cos20°+\sin20°-\sin20°}{\cos20°}\notag\\
={}&\sqrt3\notag\\
\end{align*}
\]
标签：cos,
frac,
三角函数,
变形,
beta,
sin,
notag,
alpha,
恒等
From： https://www.cnblogs.com/bxjz/p/tfid.html