■定義
\(sec \theta =\displaystyle\frac{1}{cos \theta}\) \(sech \theta =\displaystyle\frac{1}{cosh \theta}\)
\(csc \theta =\displaystyle\frac{1}{sin \theta}\) \(csch \theta =\displaystyle\frac{1}{sinh \theta}\)
\(cot \theta =\displaystyle\frac{1}{tan \theta}\) \(coth \theta =\displaystyle\frac{1}{tanh \theta}\)
■相互関係
\(sin^2 \theta + cos^2\theta=1\) \(cosh^2 \theta - sinh^2 \theta=1\)
\(sec^2 \theta - tan^2 \theta=1\) \(sech^2 \theta + tanh^2 \theta=1\)
\(csc^2 \theta - cot^2 \theta=1\) \(coth^2 \theta - csch^2 \theta=1\)
■倍角
\(cos 2\theta = cos^2 \theta - sin^2 \theta\)
\(=1-2sin^2 \theta \)
\(=2cos^2 \theta -1\)
\(tan 2\theta = \displaystyle\frac{2 tan \theta}{1-tan^2\theta}\)
\(sinh2\theta=2sinh\theta cosh\theta\)
\(cosh2\theta=cosh^2\theta +sinh^2\theta\)
\(tanh2\theta=\displaystyle\frac{2tanh\theta}{1+tanh^2\theta}\)
■２乗
\(sin^2 \theta = \displaystyle\frac{1-cos 2\theta}{2}\)
\(cos^2 \theta = \displaystyle\frac{1+cos 2\theta}{2}\)
\(sinh^2 \theta = \displaystyle\frac{cosh 2\theta-1}{2}\)
\(cosh^2 \theta = \displaystyle\frac{cosh 2\theta+1}{2}\)
■三角関数と双曲線関数の相互関係
\(cos \theta =cosh i \theta cosh \theta = cos i \theta\)
\(isin \theta = sinh i \theta isinh \theta = sin i \theta\)
■指数による定義
\(sinh \theta = \displaystyle\frac{1}{2}(e^x-e^{-x})\)
\(cosh \theta = \displaystyle\frac{1}{2}(e^x+e^{-x})\)
■ドモアブルの定理
\((cos \theta + i sin \theta )^n= e^{in\theta}=cosn \theta +i sin n \theta\)
■微分
\((sin ax)'=acos ax\) \((sec ax)'=asec ax･tan ax\)
\((cos ax)'=-asin ax\) \((csc ax)'=-acsc ax･cot ax\)
\((tan ax)'=asec^2 ax\) \((cot ax)'=-acsc^2ax\)
\((sinh ax)'=acosh ax\) \((sech ax)'=-asech ax･tanh ax\)
\((cosh ax)'=asinh ax\) \((csch ax)'=-acsch ax･coth ax\)
\((tanh ax)'=asech^2 ax\) \((coth ax)'=-acsch^2 ax\)