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$$z=\mathrm{P}^{\frac13}=e^{\frac{2\pi}3i}$$ の場合
\begin{array}{c}
+\cos_{z}x && &=& \exp_3^{(0)}x-\exp_3^{(1)}x &\leftarrow[+1,~~0~,-1]\\-\sin_{z}x &=& +\sin_{z^{-1}}x &=& \exp_3^{(1)}x-\exp_3^{(2)}x &\leftarrow[~~0~,-1,+1]\\ && -(\cos_sin_{z^{-1}}x )' &=& \exp_3^{(2)}x-\exp_3^{(0)}x &\leftarrow[-1,+1,~~0~]\\
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-\cos_{z}x && &=& -\exp_3^{(0)}x+\exp_3^{(1)}x &\leftarrow[-1,~~0~,+1]\\+\sin_{z}x &=& -\sin_{z^{-1}}x &=& -\exp_3^{(1)}x+\exp_3^{(2)}x &\leftarrow[~~0~,+1,-1]\\ && +(\cos_sin_{z^{-1}}x )' &=& -\exp_3^{(2)}x+\exp_3^{(0)}x &\leftarrow[+1,-1,~~0~]\\
\end{array}
$$z=\mathrm{P}^{\frac14}=e^{\frac{2\pi}4i}=e^{\frac{\pi}2i}$$ の場合
\begin{array}{c}
+\cos x &=& +\cos_zx &=& +\cos_{z^{-1}}x &=& \exp_4^{(0)}x-\exp_4^{(2)}x &\leftarrow[+1,~~0~,-1,~~0~]\\-\sin x &=& -\sin_zx &=& +\sin_{z^{-1}}x &=& \exp_4^{(1)}x-\exp_4^{(3)}x &\leftarrow[~~0~,-1,~~0~,+1]\\-\cos x &=& -\cos_zx &=& -\cos_{z^{-1}}x &=& \exp_4^{(2)}x-\exp_4^{(0)}x &\leftarrow[-1,~~0~,+1,~~0~]\\+\sin x &=& +\sin_zx &=& -\sin_{z^{-1}}x &=& \exp_4^{(3)}x-\exp_4^{(1)}x &\leftarrow[~~0~,+1,~~0~,-1]\\
\end{array}
$$z=\mathrm{P}^{\frac15}=e^{\frac{2\pi}5i}$$ の場合（$$\phi'=\phi^{-1}=\frac{\sqrt5-1}2$$）
\begin{array}{c}
+\cos_zx && &=& +\exp_5^{(0)}x-\exp_5^{(3)}x-\phi'\exp_5^{(2)}x+\phi'\exp_5^{(1)}x &\leftarrow[+1~,~~0~~,-1~,-\phi',+\phi']\\-\sin_zx &=& +\sin_{z^{-1}}x &=& +\exp_5^{(1)}x-\exp_5^{(4)}x-\phi'\exp_5^{(3)}x+\phi'\exp_5^{(2)}x &\leftarrow[~~0~~,-1~,-\phi',+\phi',+1~]\\ && -(\cos_{z^{-1}}x sin_zx)' &=& +\exp_5^{(2)}x-\exp_5^{(5)}x-\phi'\exp_5^{(4)}x+\phi'\exp_5^{(3)}x &\leftarrow[-1~,-\phi',+\phi',+1~,~~0~~]\\ && -(+\sin_{z^{-1}}xsin_zx)'' &=& +\exp_5^{(3)}x-\exp_5^{(0)}x-\phi'\exp_5^{(5)}x+\phi'\exp_5^{(4)}x &\leftarrow[-\phi',+\phi',+1~,~~0~~,-1~]\\ && -(-\cos_{z^{-1}}xsin_zx)''' &=& +\exp_5^{(4)}x-\exp_5^{(1)}x-\phi'\exp_5^{(0)}x+\phi'\exp_5^{(5)}x &\leftarrow[+\phi',+1~,~~0~~,-1~,-\phi']\\
\\
-\cos_zx && &=& -\exp_5^{(0)}x+\exp_5^{(3)}x+\phi'\exp_5^{(2)}x-\phi'\exp_5^{(1)}x &\leftarrow[-1~,~~0~~,+1~,+\phi',-\phi']\\+\sin_zx &=& -\sin_{z^{-1}}x &=& -\exp_5^{(1)}x+\exp_5^{(4)}x+\phi'\exp_5^{(3)}x-\phi'\exp_5^{(2)}x &\leftarrow[~~0~~,+1~,+\phi',-\phi',-1~]\\ && +(\cos_{z^{-1}}x sin_zx)' &=& -\exp_5^{(2)}x+\exp_5^{(0)}x+\phi'\exp_5^{(4)}x-\phi'\exp_5^{(3)}x &\leftarrow[+1~,+\phi',-\phi',-1~,~~0~~]\\ && +(-\sin_{z^{-1}}xsin_zx)'' &=& -\exp_5^{(3)}x+\exp_5^{(1)}x+\phi'\exp_5^{(0)}x-\phi'\exp_5^{(4)}x &\leftarrow[+\phi',-\phi',-1~,~~0~~,+1~]\\ && +(+\cos_{z^{-1}}xsin_zx)''' &=& -\exp_5^{(4)}x+\exp_5^{(2)}x+\phi'\exp_5^{(1)}x-\phi'\exp_5^{(0)}x &\leftarrow[-\phi',-1~,~~0~~,+1~,+\phi']\\
\end{array}
$$z=\mathrm{P}^{\frac16}=e^{\frac{2\pi}6i}=e^{\frac{\pi}3i}$$ の場合
\begin{array}{c}
+\cos_{z}x && &=& \exp_6^{(0)}x-\exp_6^{(4)}x-\exp_6^{(3)}x+\exp_6^{(1)}x &\leftarrow[+1,~~0~,-1,-1,~~0~,+1]\\-\sin_{z}x &=& +\sin_{z^{-1}}x &=& \exp_6^{(1)}x-\exp_6^{(5)}x-\exp_6^{(4)}x+\exp_6^{(2)}x &\leftarrow[~~0~,-1,-1,~~0~,+1,+1]\\ && -(\cos_sin_{z^{-1}}x )' &=& \exp_6^{(2)}x-\exp_6^{(0)}x-\exp_6^{(5)}x+\exp_6^{(3)}x &\leftarrow[-1,-1,~~0~,+1,+1,~~0~]\\\\-\cos_{z}x && &=& \exp_6^{(3)}x-\exp_6^{(1)}x-\exp_6^{(0)}x+\exp_6^{(4)}x &\leftarrow[-1,~~0~,+1,+1,~~0~,-1]\\+\sin_{z}x &=& -\sin_{z^{-1}}x &=& \exp_6^{(4)}x-\exp_6^{(2)}x-\exp_6^{(1)}x+\exp_6^{(5)}x &\leftarrow[~~0~,+1,+1,~~0~,-1,-1]\\ && +(\cos_sin_{z^{-1}}x )' &=& \exp_6^{(5)}x-\exp_6^{(3)}x-\exp_6^{(2)}x+\exp_6^{(0)}x &\leftarrow[+1,+1,~~0~,-1,-1,~~0~]\\
\end{array}
