逆三角関数と逆双曲線関数の負角

逆三角関数の負角

(1)

\[ \Sin^{\circ}\left(-z\right)=-\Sin^{\circ}z \]

(2)

\[ \Cos^{\circ}\left(-z\right)=-\Cos^{\circ}\left(z\right)+\pi \]

(3)

\[ \Tan^{\circ}\left(-z\right)=-\Tan^{\circ}z \]

(4)

\[ \Sin^{-1,\circ}\left(-z\right)=-\Sin^{-1,\circ}z \]

(5)

\[ \Cos^{-1,\circ}\left(-z\right)=-\Cos^{-1,\circ}z+\pi \]

(6)

\[ \Tan^{-1,\circ}\left(-z\right)=-\Tan^{-1,\circ}z \]

(1)

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

(1)-2

\begin{align*} \Sin^{\circ}\left(-z\right) & =-i\Log\left(-iz+\sqrt{1-\left(-z\right)^{2}}\right)\\ & =-i\Log\frac{\left(-iz+\sqrt{1-z^{2}}\right)\left(iz+\sqrt{1-z^{2}}\right)}{\left(iz+\sqrt{1-z^{2}}\right)}\\ & =-i\Log\left(iz+\sqrt{1-z^{2}}\right)^{-1}\\ & =-i\left\{ -\Log\left(iz+\sqrt{1-z^{2}}\right)+2\pi i\delta_{\pi,\Arg\left(iz+\sqrt{1-z^{2}}\right)}\right\} \\ & =i\Log\left(iz+\sqrt{1-z^{2}}\right)\\ & =-\Sin^{\circ}z \end{align*}

(2)

\begin{align*} \Cos^{\circ}\left(-z\right) & =-\Sin^{\circ}\left(-z\right)+\frac{\pi}{2}\\ & =\Sin^{\circ}\left(z\right)+\frac{\pi}{2}\\ & =-\Cos^{\circ}\left(z\right)+\pi \end{align*}

(3)

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

(3)-2

\begin{align*} \Tan^{\circ}\left(-z\right) & =\frac{i}{2}\left(\Log\left(1-i\left(-z\right)\right)-\Log\left(1+i\left(-z\right)\right)\right)\\ & =\frac{i}{2}\left(\Log\left(1+iz\right)-\Log\left(1-iz\right)\right)\\ & =-\frac{i}{2}\left(\Log\left(1-iz\right)-\Log\left(1+iz\right)\right)\\ & =-\Tan\left(z\right) \end{align*}

(4)

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

(4)-2

\begin{align*} \Sin^{-1,\circ}\left(-z\right) & =\Sin^{\circ}\left(-\frac{1}{z}\right)\\ & =-\Sin^{\circ}\frac{1}{z}\\ & =-\Sin^{-1,\circ}z \end{align*}

(5)

\begin{align*} \Cos^{-1,\circ}\left(-z\right) & =\Cos^{\circ}\left(-\frac{1}{z}\right)\\ & =-\Cos^{\circ}\left(\frac{1}{z}\right)+\pi\\ & =-\Cos^{-1,\circ}z+\pi \end{align*}

(6)

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

(6)-2

\begin{align*} \Tan^{-1,\circ}\left(-z\right) & =\Tan^{\circ}\left(-\frac{1}{z}\right)\\ & =-\Tan^{\circ}\frac{1}{z}\\ & =-\Tan^{-1,\circ}z \end{align*}

逆双曲線関数の負角

(1)

\[ \Sinh^{\circ}\left(-z\right)=-\Sinh^{\circ}z \]

(2)

\[ \Tanh^{\circ}\left(-z\right)=-\Tanh^{\circ}z \]

(3)

\[ \Sinh^{-1,\circ}\left(-z\right)=-\Sinh^{-1,\circ}z \]

(4)

\[ \Tanh^{-1,\circ}\left(-z\right)=-\Tanh^{-1,\circ}z \]

(1)

\begin{align*} \Sinh^{\circ}\left(-z\right) & =\Sinh^{\circ}\left(-\sinh\Sinh^{\circ}z\right)\\ & =\Sinh^{\circ}\sinh\left(-\Sinh^{\circ}z\right)\\ & =-\Sinh^{\circ}z \end{align*}

(1)-2

\begin{align*} \sinh^{\circ}\left(-z\right) & =\Log\left(-z+\sqrt{1+\left(-z\right)^{2}}\right)\\ & =\Log\frac{\left(-z+\sqrt{1+z^{2}}\right)\left(z+\sqrt{1+z^{2}}\right)}{\left(z+\sqrt{1+z^{2}}\right)}\\ & =\Log\left(z+\sqrt{1+z^{2}}\right)^{-1}\\ & =-\Log\left(z+\sqrt{1+z^{2}}\right)+2\pi i\delta_{\pi,\Arg\left(z+\sqrt{1+z^{2}}\right)}\\ & =-\Log\left(z+\sqrt{1+z^{2}}\right)\\ & =-\sinh^{\circ}z \end{align*}

(2)

\begin{align*} \Tanh^{\circ}\left(-z\right) & =\Tanh^{\circ}\left(-\tanh\Tanh^{\circ}z\right)\\ & =\Tanh^{\circ}\tanh\left(-\Tanh^{\circ}z\right)\\ & =-\Tanh^{\circ}z \end{align*}

(2)-2

\begin{align*} \Tanh^{\circ}\left(-z\right) & =\frac{1}{2}\left(\Log\left(1+\left(-z\right)\right)-\Log\left(1-\left(-z\right)\right)\right)\\ & =\frac{1}{2}\left(\Log\left(1-z\right)-\Log\left(1+z\right)\right)\\ & =-\frac{1}{2}\left(\Log\left(1+z\right)-\Log\left(1-z\right)\right)\\ & =-\Tanh^{\circ}z \end{align*}

(3)

\begin{align*} \Sinh^{-1,\circ}\left(-z\right) & =\Sinh^{-1,\circ}\left(-\sinh^{-1}\Sinh^{-1,\circ}z\right)\\ & =\Sinh^{-1,\circ}\sinh^{-1}\left(-\Sinh^{-1,\circ}z\right)\\ & =-\Sinh^{-1,\circ}z \end{align*}

(3)-2

\begin{align*} \Sinh^{-1,\circ}\left(-z\right) & =\Sinh^{\circ}\left(-\frac{1}{z}\right)\\ & =-\Sinh^{\circ}\frac{1}{z}\\ & =-\Sinh^{-1,\circ}z \end{align*}

(4)

\begin{align*} \Tanh^{-1,\circ}\left(-z\right) & =\Tanh^{-1,\circ}\left(-\tanh^{-1}\Tanh^{-1,\circ}z\right)\\ & =\Tanh^{-1,\circ}\tanh^{-1}\left(-\Tanh^{-1,\circ}z\right)\\ & =-\Tanh^{-1,\circ}z \end{align*}

(4)-2

\begin{align*} \Tanh^{-1,\circ}\left(-z\right) & =\Tanh^{\circ}\left(-\frac{1}{z}\right)\\ & =-\Tanh^{\circ}\frac{1}{z}\\ & =-\Tanh^{-1,\circ}z \end{align*}

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逆三角関数と逆双曲線関数の負角

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