三角関数の公式_オイラーの公式を使って

べき乗

二乗

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {sin^2 \ \theta }}
&=& \frac{e^{i \theta } -e^{-i \theta }}{2i} \frac{e^{i \theta } -e^{-i \theta }}{2i}\\
&=& -\frac{1}{4} \left[ e^{2i \theta } - 1 - 1 + e^{2i \theta } \right] \\
&=& \frac{1}{2} - \frac{cos \ 2 \theta}{2} \\
&=& \boldsymbol{ \LARGE {\frac{1 - cos \ 2 \theta}{2} }}
\end{eqnarray} \)

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {cos^2 \ \theta }}
&=& \frac{e^{i \theta } +e^{-i \theta }}{2} \frac{e^{i \theta } +e^{-i \theta }}{2}\\
&=& \frac{1}{4} \left[ e^{2i \theta } + 1 + 1 + e^{2i \theta } \right] \\
&=& \frac{1}{2} + \frac{cos \ 2 \theta}{2} \\
&=& \boldsymbol{ \LARGE {\frac{1 + cos \ 2 \theta}{2} }}
\end{eqnarray} \)

三乗

3倍角の公式から，

\( \Large \displaystyle sin \ (3 \theta) =3 \ sin \theta -4 \ sin^3 \theta \)

\( \Large \displaystyle \boldsymbol{ \LARGE {sin^3 \theta = \frac{ 3 \ sin \theta - sin \ (3 \theta) }{4}}} \)

\( \Large \displaystyle cos \ (3 \theta)= 4 \ cos^3 \theta - 3 \ cos \theta \)

\( \Large \displaystyle \boldsymbol{ \LARGE {cos^3 \theta = \frac{ 3 \ cos \theta + cos \ (3 \theta) }{4}}} \)

四乗

\( \Large \displaystyle sin^2 \ \theta = \frac{1 - cos \ 2 \theta}{2} \)

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {sin^4 \ \theta }}
&=& \left( \frac{1 - cos \ 2 \theta}{2} \right)^2 \\
&=& \frac{1}{4} \left( 1 - 2 \ cos \ 2 \theta + cos^2 2 \theta \right) \\
&=& \frac{1}{4} \left( 1 - 2 \ cos \ 2 \theta + \frac{ 1+ cos 4 \theta}{2} \right) \\
&=& \boldsymbol{ \LARGE {\frac{3 - 4 \ cos \ 2 \theta+ cos \ 4 \theta}{8} }}
\end{eqnarray} \)

\( \Large \displaystyle cos^2 \ \theta = \frac{1 + cos \ 2 \theta}{2} \)

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {cos^4 \ \theta }}
&=& \left( \frac{1 + cos \ 2 \theta}{2} \right)^2 \\
&=& \frac{1}{4} \left( 1 + 2 \ cos \ 2 \theta + cos^2 2 \theta \right) \\
&=& \frac{1}{4} \left( 1 + 2 \ cos \ 2 \theta + \frac{ 1+ cos 4 \theta}{2} \right) \\
&=& \boldsymbol{ \LARGE {\frac{3 + 4 \ cos \ 2 \theta+ cos \ 4 \theta}{8} }}
\end{eqnarray} \)

次は，その他，