逆三角関数の負角、余角、逆数

逆三角関数の負角

(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

(1)

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

(1)-2

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

(2)

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

(3)

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

(3)-2

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

(4)

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

(4)-2

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

(5)

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

(6)

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

(6)-2

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

逆三角関数の余角

(1)

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

(2)

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

(3)

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

(1)

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

(2)

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

(3)

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

逆三角関数の逆数

(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

(1)〜(6)

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

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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