逆三角関数の負角、余角、逆数

逆三角関数の負角

(1)

\[ \sin^{\circ}\left(-x\right)=-\sin^{\circ}x \]

(2)

\[ \cos^{\circ}\left(-x\right)=\pi-\cos^{\circ}x \]

(3)

\[ \tan{}^{\circ}\left(-x\right)=-\tan^{\circ}x \]

(4)

\[ \sin^{-1,\circ}\left(-x\right)=-\sin^{-1,\circ}x \]

(5)

\[ \cos^{-1,\circ}\left(-x\right)=\pi-\cos^{-1,\circ}x \]

(6)

\[ \tan{}^{-1,\circ}\left(-x\right)=-\tan^{-1,\circ}x \]

(1)

奇関数の逆関数は奇関数なので、
\[ \sin^{\circ}\left(-x\right)=-\sin^{\circ}x \]

(1)-2

\begin{align*} \sin^{\circ}\left(-x\right) & =\sin^{\circ}\left(-\sin\left(\sin^{\circ}x\right)\right)\\ & =\sin^{\circ}\left(\sin(-\sin^{\circ}x)\right)\\ & =-\sin^{\circ}x \end{align*}

(2)

\begin{align*} \cos^{\circ}\left(-x\right) & =\cos^{\circ}\left(-\cos\left(\cos^{\circ}x\right)\right)\\ & =\cos^{\circ}\left(\cos(\pi-\cos^{\circ}x)\right)\\ & =\pi-\cos^{\circ}x \end{align*}

(3)

奇関数の逆関数は奇関数なので、
\[ \tan{}^{\circ}\left(-x\right)=-\tan^{\circ}x \]

(3)-2

\begin{align*} \tan^{\circ}\left(-x\right) & =\tan^{\circ}\left(-\tan\left(\tan^{\circ}x\right)\right)\\ & =\tan^{\circ}\left(\tan(-\tan^{\circ}x)\right)\\ & =\tan^{\circ}x \end{align*}

(4)

奇関数の逆関数は奇関数なので、
\[ \sin^{-1,\circ}\left(-x\right)=-\sin^{-1,\circ}x \]

(4)-2

\begin{align*} \sin^{-1,\circ}\left(-x\right) & =\sin^{-1,\circ}\left(-\sin^{-1}\left(\sin^{-1,\circ}x\right)\right)\\ & =\sin^{-1,\circ}\left(\sin^{-1}(-\sin^{-1\circ}x)\right)\\ & =-\sin^{-1,\circ}x \end{align*}

(5)

\begin{align*} \cos^{-1,\circ}\left(-x\right) & =\cos^{-1,\circ}\left(-\cos^{-1}\left(\cos^{-1,\circ}x\right)\right)\\ & =\cos^{-1,\circ}\left(\cos^{-1}(\pi-\cos^{-1,\circ}x)\right)\\ & =\pi-\cos^{-1,\circ}x \end{align*}

(6)

奇関数の逆関数は奇関数なので、
\[ \tan{}^{-1,\circ}\left(-x\right)=-\tan^{-1,\circ}x \]

(6)-2

\begin{align*} \tan^{-1,\circ}\left(-x\right) & =\tan^{-1,\circ}\left(-\tan^{-1}\left(\tan^{-1,\circ}x\right)\right)\\ & =\tan^{-1,\circ}\left(\tan(-\tan^{-1,\circ}x)\right)\\ & =\tan^{-1,\circ}x \end{align*}

逆三角関数の余角

(1)

\[ \cos^{\circ}x+\sin^{\circ}x=\frac{\pi}{2} \]

(2)

\[ \tan^{\circ}x+\tan^{-1,\circ}x=\frac{\pi}{2} \]

(3)

\[ \sin^{-1,\circ}x+\cos^{-1,\circ}x=\frac{\pi}{2} \]

(1)

\begin{align*} \frac{\pi}{2} & =x+\frac{\pi}{2}-x\\ & =\cos^{\circ}\cos x+\sin^{\circ}\sin\left(\frac{\pi}{2}-x\right)\\ & =\cos^{\circ}y+\sin^{\circ}y\qquad,\qquad y=\cos x \end{align*}

(2)

\begin{align*} \frac{\pi}{2} & =x+\frac{\pi}{2}-x\\ & =\tan^{\circ}\tan x+\tan^{-1,\circ}\tan^{-1}\left(\frac{\pi}{2}-x\right)\\ & =\tan^{\circ}y+\tan^{-1,\circ}y\qquad,\qquad y=\tan x \end{align*}

(3)

\begin{align*} \frac{\pi}{2} & =x+\frac{\pi}{2}-x\\ & =\sin^{-1,\circ}\sin^{-1}x+\cos^{-1,\circ}\cos^{-1}\left(\frac{\pi}{2}-x\right)\\ & =\sin^{-1,\circ}y+\cos^{-1,\circ}y\qquad,\qquad y=\sin^{-1}x \end{align*}

逆三角関数の逆数

(1)

\[ \sin^{\circ}\left(\frac{1}{x}\right)=\sin^{-1,\circ}\left(x\right) \]

(2)

\[ \cos^{\circ}\left(\frac{1}{x}\right)=\cos^{-1,\circ}\left(x\right) \]

(3)

\[ \tan^{\circ}\left(\frac{1}{x}\right)=\tan^{-1,\circ}\left(x\right) \]

(4)

\[ \sin^{-1,\circ}\left(\frac{1}{x}\right)=\sin^{\circ}\left(x\right) \]

(5)

\[ \cos^{-1,\circ}\left(\frac{1}{x}\right)=\cos^{\circ}\left(x\right) \]

(6)

\[ \tan^{-1,\circ}\left(\frac{1}{x}\right)=\tan^{\circ}\left(\frac{1}{x}\right) \]

(1)〜(6)

\[ f\left(\frac{1}{x}\right)=f^{\circ,-1,\circ}(x) \]

より成り立つ。

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逆三角関数の負角、余角、逆数

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