よく使う三角関数関係の公式をこのページにメモしておく。
基本公式
\begin{align} &\sin x = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} x^{2n+1} \\[5pt] &\cos x = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n)!} x^{2n} \\[5pt] &\Atan\,x = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} x^{2n+1} \4 ( |x| \lt 1 ) \\[5pt] &\frac{1}{\cos^{2}x} = 1 + \tan^{2} x \end{align}
加法定理
\begin{align} &\sin( x \pm y ) = \sin x \cos y \pm \cos x \sin y \\[5pt] &\cos ( x \pm y ) = \cos x \cos y \mp \sin x \sin y \\[5pt] &\tan ( x \pm y ) = \frac{\tan x\pm\tan y}{1\mp\tan x\tan y} \end{align}
倍角公式
\begin{align} &\sin 2x = 2 \sin x \cos x \\[5pt] &\cos 2x = \cos^{2} x - \sin^{2} x \notag \\ &\5\ = 2 \cos^{2} x - 1 \\ &\5\ = 1 - 2 \sin^{2} x \notag \\[5pt] &\tan 2x = \frac{2\tan x}{1-\tan^{2}x} \\[10pt] &\sin 3x = 3 \sin x - 4 \sin^{3} x \\[5pt] &\cos 3x = 4 \cos^{3} x - 3 \cos x \\[5pt] &\tan 3x = \frac{3\tan x-\tan^{3}x}{1-3\tan^{2}x} \end{align}
半角公式
\begin{align} &\sin^{2} \Bigl( \frac{x}{2} \Bigr) = \frac{1}{2} ( 1 - \cos x ) \\[5pt] &\cos^{2} \Bigl( \frac{x}{2} \Bigr) = \frac{1}{2} ( 1 + \cos x ) \\[5pt] &\tan \Bigl( \frac{x}{2} \Bigr) = \frac{1-\cos x}{\sin x} = \frac{\sin x}{1+\cos x} \end{align}
和積公式
\begin{align} &\sin x + \sin y = 2 \sin \Bigl( \frac{x+y}{2} \Bigr) \cos \Bigl( \frac{x-y}{2} \Bigr) \\[5pt] &\sin x - \sin y = 2 \cos \Bigl( \frac{x+y}{2} \Bigr) \sin \Bigl( \frac{x-y}{2} \Bigr) \\[5pt] &\cos x + \cos y = 2 \cos \Bigl( \frac{x+y}{2} \Bigr) \cos \Bigl( \frac{x-y}{2} \Bigr) \\[5pt] &\cos x - \cos y = -2 \sin \Bigl( \frac{x+y}{2} \Bigr) \sin \Bigl( \frac{x-y}{2} \Bigr) \end{align}
積和公式
\begin{align} &\sin x \sin y = \frac{1}{2} \bigl( -\cos ( x + y ) + \cos ( x - y ) \bigr) \\[5pt] &\cos x \cos y = \frac{1}{2} \bigl( \cos ( x + y ) + \cos ( x - y ) \bigr) \\[5pt] &\sin x \cos y = \frac{1}{2} \bigl( \sin ( x + y ) + \sin ( x - y ) \bigr) \\[5pt] &\cos x \sin y = \frac{1}{2} \bigl( \sin ( x + y ) - \sin ( x - y ) \bigr) \end{align}
不定積分
\begin{align} &\int \sin^{2} x \, dx = \frac{x}{2} - \frac{1}{4} \sin 2x + C \\[5pt] &\int \cos^{2} x \, dx = \frac{x}{2} + \frac{1}{4} \sin 2x + C \\[5pt] &\int x \sin x \, dx = -x \cos x + \sin x + C \\[5pt] &\int x \cos x \, dx = x \sin x + \cos x + C \\[5pt] &\int x^{2} \sin x \, dx = -x^{2} \cos x + 2x \sin x + 2\cos x + C \\[5pt] &\int x^{2}\cos x \, dx = x^{2} \sin x + 2x \cos x - 2\sin x + C \end{align}
定積分
\begin{align} &\int_{0}^{\textstyle \frac{\pi}{2}} \sin x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos x \, dx = 1 \\[5pt] &\int_{0}^{\textstyle \frac{\pi}{2}} \sin^{2} x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos^{2} x \, dx = \frac{\pi}{4} \\[5pt] &\int_{0}^{\textstyle \frac{\pi}{2}} \sin^{3} x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos^{3} x \, dx = \frac{2}{3} \\[5pt] &\int_{0}^{\textstyle \frac{\pi}{2}} \sin^{4} x \, dx = \int_{0}^{\textstyle \frac{\pi}{2}} \cos^{4} x \, dx = \frac{3\pi}{16} \end{align}