三角関数の還元(負角・余角・補角)公式
三角関数の還元公式
負角
(1)
\[ \sin(-x)=-\sin x \]
(2)
\[ \cos(-x)=\cos x \]
(3)
\[ \tan(-x)=-\tan x \]
余角
(4)
\[ \sin\left(\frac{\pi}{2}-x\right)=\cos x \]
(5)
\[ \cos\left(\frac{\pi}{2}-x\right)=\sin x \]
(6)
\[ \tan\left(\frac{\pi}{2}-x\right)=\cot x \]
補角
(7)
\[ \sin\left(\pi-x\right)=\sin x \]
(8)
\[ \cos\left(\pi-x\right)=-\cos x \]
(9)
\[ \tan\left(\pi-x\right)=-\tan x \]
(1)
\begin{align*} \sin(-x) & =\frac{e^{-ix}-e^{ix}}{2i}\\ & =-\frac{e^{ix}-e^{-ix}}{2i}\\ & =-\sin x \end{align*}
(2)
\begin{align*} \cos(-x) & =\frac{e^{-ix}+e^{ix}}{2}\\ & =\cos x \end{align*}
(3)
\begin{align*} \tan(-x) & =\frac{\sin(-x)}{\cos(-x)}\\ & =\frac{-\sin(x)}{\cos x}\\ & =-\tan x \end{align*}
(4)
\begin{align*} \sin\left(\frac{\pi}{2}-x\right) & =\sin\frac{\pi}{2}\cos x-\cos\frac{\pi}{2}\sin x\\ & =\cos x \end{align*}
(5)
\begin{align*} \cos\left(\frac{\pi}{2}-x\right) & =\cos\frac{\pi}{2}\cos x+\sin\frac{\pi}{2}\sin x\\ & =\sin x \end{align*}
(6)
\begin{align*} \tan\left(\frac{\pi}{2}-x\right) & =\frac{\sin\left(\frac{\pi}{2}-x\right)}{\cos\left(\frac{\pi}{2}-x\right)}\\ & =\frac{\cos x}{\sin x}\\ & =\cot x \end{align*}
(7)
\begin{align*} \sin\left(\pi-x\right) & =\sin\pi\cos x-\cos\pi\sin x\\ & =\sin x \end{align*}
(8)
\begin{align*} \cos\left(\pi-x\right) & =\cos\pi\cos x+\sin\pi\sin x\\ & =-\cos x \end{align*}
(9)
\begin{align*} \tan\left(\pi-x\right) & =\frac{\sin(\pi-x)}{\cos(\pi-x)}\\ & =\frac{\sin x}{-\cos x}\\ & =-\tan x \end{align*}
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