ピタゴラスの基本三角関数公式
ピタゴラスの基本三角関数公式
(1)
\[ \cos^{2}x+\sin^{2}x=1 \]
(2)
\[ 1+\tan^{2}x=\cos^{-2}x \]
(3)
\[ 1+\cot^{2}x=\sin^{-2}x \]
(1)
\begin{align*} \cos^{2}x+\sin^{2}x & =\left(\cos x+i\sin x\right)\left(\cos x-i\sin x\right)\\ & =e^{ix}e^{-ix}\\ & =1 \end{align*}
(2)
\begin{align*} 1+\tan^{2}x & =\cos^{-2}x(\cos^{2}x+\sin^{2}x)\\ & =\cos^{-2}x \end{align*}
(3)
\begin{align*} 1+\cot^{2}x & =\sin^{-2}x(\sin^{2}x+\cos^{2}x)\\ & =\sin^{-2}x \end{align*}
基本双曲線関数公式
(1)
\[ \cosh^{2}x-\sinh^{2}x=1 \]
(2)
\begin{align*} 1-\tanh^{2}x & =\cosh^{-2}x \end{align*}
(3)
\[ 1-\coth^{2}x=-\sinh^{-2}x \]
(1)
\begin{align*} 1 & =\cos^{2}ix+\sin^{2}ix\\ & =\cosh^{2}x-\sinh^{2}ix \end{align*}
(2)
\begin{align*} 1-\tanh^{2}x & =\cosh^{-2}x(\cosh^{2}x-\sinh^{2}x)\\ & =\cosh^{-2}x \end{align*}
(3)
\begin{align*} 1-\coth^{2}x & =\sinh^{-2}x(\sinh^{2}x-\cosh^{2}x)\\ & =-\sinh^{-2}x \end{align*}
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タイトル | ピタゴラスの基本三角関数公式 |
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SNSボタン |
- 三角関数と双曲線関数の実部と虚部\[ \tan z=\frac{\sin\left(2\Re z\right)+i\sinh\left(2\Im z\right)}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \]
- 三角関数と双曲線関数の対数\[ \log\sin x=-\log2+\frac{\pi}{2}i-ix-Li_{1}\left(e^{2ix}\right) \]
- 三角関数と双曲線関数の積分\[ \int\cos xdx=\sin x \]
- 三角関数と双曲線関数のn乗積分\[ \int\sin^{2n+m_{\pm}}xdx=\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1+m_{\pm}}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sin^{m_{\pm}}xdx\right\} \]
- 三角関数と双曲線関数の冪乗積分漸化式\[ \int\sin^{n}xdx=-\frac{1}{n}\cos x\sin^{n-1}x+\frac{n-1}{n}\int\sin^{n-2}xdx\qquad(n\ne0) \]