偏角の3角関数

by nomura · 2021年6月1日

偏角の3角関数

(1)

\sin Arg z = \frac{ℑ z}{| z |}

\cos Arg z = \frac{ℜ z}{| z |}

\tan Arg z = \frac{ℑ z}{ℜ z}

\begin{aligned} \sin Arg z & = \frac{e^{i Arg z} - e^{- i Arg z}}{2 i} \\ = \frac{sgn z - {sgn}^{- 1} z}{2 i} \\ = \frac{1}{2 i} (\frac{z}{| z |} - \frac{| z |}{z}) \\ = \frac{1}{2 i} (\frac{z | z | - \overset{―}{z} | z |}{{| z |}^{2}}) \\ = \frac{ℑ z}{| z |} \end{aligned}

\begin{aligned} \cos Arg z & = \frac{e^{i Arg z} + e^{- i Arg z}}{2} \\ = \frac{sgn z + {sgn}^{- 1} z}{2} \\ = \frac{1}{2} (\frac{z}{| z |} + \frac{| z |}{z}) \\ = \frac{1}{2} (\frac{z | z | + \overset{―}{z} | z |}{{| z |}^{2}}) \\ = \frac{ℜ z}{| z |} \end{aligned}

\begin{aligned} \tan Arg z & = \frac{\sin Arg z}{\cos Arg z} \\ = \frac{ℑ z}{ℜ z} \end{aligned}

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