三角関数の公式_オイラーの公式を使って

加法定理

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {sin \ (x \pm y) }}
&=& Im \left[ e^{i(x \pm y)} \right] \\
&=& Im \left[ e^{ix} \cdot e^{ \pm iy} \right] \\
&=& Im \left[ ( cos \ x +i \ sin \ x) \times \{ cos ( \pm y) + i \ sin ( \pm y) \} \right] \\
&=& Im \left[ ( cos \ x \cdot cos \ y \mp sin \ x \cdot sin \ y ) + i( sin \ x \cdot cos \ y \pm cos \ x \cdot sin \ y ) \right] \\
&=& \boldsymbol{ \LARGE {sin \ x \cdot cos \ y \pm cos \ x \cdot sin \ y }}
\end{eqnarray} \)

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {cos \ (x \pm y) }}
&=& Re \left[ e^{i(x \pm y)} \right] \\
&=& Re \left[ e^{ix} \cdot e^{ \pm iy} \right] \\
&=& Re \left[ ( cos \ x +i \ sin \ x) \times \{ cos ( \pm y) + i \ sin ( \pm y) \} \right] \\
&=& Re \left[ ( cos \ x \cdot cos \ y \mp sin \ x \cdot sin \ y ) + i( sin \ x \cdot cos \ y \pm cos \ x \cdot sin \ y ) \right] \\
&=&\boldsymbol{ \LARGE {cos \ x \cdot cos \ y \mp sin \ x \cdot sin \ y}}
\end{eqnarray} \)

倍角の公式

二倍角

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

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

三倍角

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {sin \ (3 \theta) }}
&=& Im \left[ e^{3 i \theta)} \right] \\
&=& Im \left[ e^{i \theta} \cdot e^{ i \theta} \cdot e^{ i \theta} \right] \\
&=& Im \left[ ( cos \ \theta +i \ sin \ \theta) ^3 \right] \\
&=& Im \left[ ( cos^3 \theta + 3 i \ sin \theta \ cos^2 \theta - 3 \ sin^2 \theta \ cos \theta- i \ sin^3 \theta ) \right] \\
&=& 3 \ sin \theta \ cos^2 \theta - sin^3 \theta \\
&=& 3 \ sin \theta \ cos^2 \theta - sin^3 \theta +3 \ sin^2 \theta \ sin \theta - 3 \ sin^2 \theta \ sin \theta \\
&=& 3 \ sin \theta \ (sin^2 \theta + cos^2 \theta) - sin^3 \theta - 3 \ sin^3 \theta \\
&=& \boldsymbol{ \LARGE {3 \ sin \theta -4 \ sin^3 \theta }}
\end{eqnarray} \)

\( \Large \displaystyle \begin{eqnarray} \boldsymbol{ \LARGE {cos \ (3 \theta) }}
&=& Re \left[ e^{3 i \theta)} \right] \\
&=& Re \left[ e^{i \theta} \cdot e^{ i \theta} \cdot e^{ i \theta} \right] \\
&=& Re \left[ ( cos \ \theta +i \ sin \ \theta) ^3 \right] \\
&=& Re \left[ ( cos^3 \theta + 3 i \ sin \theta \ cos^2 \theta - 3 \ sin^2 \theta \ cos \theta- i \ sin^3 \theta ) \right] \\
&=& cos^3 \theta - 3 \ sin^2 \theta \ cos \theta \\
&=& cos^3 \theta - 3 \ sin^2 \theta \ cos \theta - 3 \ cos^2 \theta \ cos \theta + 3 \ cos^2 \theta \ cos \theta \\
&=& cos^3 \theta - 3 \ (sin^2 \theta + cos^2 \theta) \ cos \theta - 3 \ cos^2 \theta \ cos \theta + 3 \ cos^3 \\
&=& \boldsymbol{ \LARGE {4 \ cos^3 \theta - 3 \ cos \theta}}
\end{eqnarray} \)

次は，和積公式，