91Ó°ÊÓ

The numerical values of the sine and cosine function at $\frac{\mathbf{Ï€}}{\mathbf{2}}$, and $\mathbf{Ï€}$are

$$\begin{gathered} \mathit{s}\mathit{i}\mathit{n}\left(\frac{\mathrm{Ï€}}{2}\right)\mathbf{=}\mathbf{1} \\ \mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathrm{Ï€}}{2}\right)\mathbf{=}\mathbf{0} \\ \mathit{s}\mathit{i}\mathit{n}\mathbf{}\left(\mathrm{Ï€}\right)\mathbf{=}\mathbf{0} \\ \mathit{c}\mathit{o}\mathit{s}\mathbf{}\left(\mathrm{Ï€}\right)\mathbf{=}\mathbf{-}\mathbf{1} \end{gathered}$$

The result (6.14) is $\left(\begin{array}{cc}\cos \mathrm{ϕ}& -\sin \mathrm{ϕ}\\ \sin \mathrm{ϕ}& \cos \mathrm{ϕ}\end{array}\right)\left(\begin{array}{cc}\cos \mathrm{θ}& -\sin \mathrm{θ}\\ \sin \mathrm{θ}& \cos \mathrm{θ}\end{array}\right)=\left(\begin{array}{cc}\cos (\mathrm{θ}+\mathrm{ϕ})& -\sin (\mathrm{θ}+\mathrm{ϕ})\\ \sin (\mathrm{θ}+\mathrm{ϕ})& \cos (\mathrm{θ}+\mathrm{ϕ})\end{array}\right)$.

It needs to be verified for $\mathrm{θ}=\mathrm{ϕ}=\mathrm{π}/2$.

Substitute into the left-hand side of the relation $(6.14)$and evaluate the result.

$\left(\begin{array}{cc}\cos \frac{\mathrm{Ï€}}{2}& -\sin \frac{\mathrm{Ï€}}{2}\\ \sin \frac{\mathrm{Ï€}}{2}& \cos \frac{\mathrm{Ï€}}{2}\end{array}\right)\left(\begin{array}{cc}\cos \frac{\mathrm{Ï€}}{2}& -\sin \frac{\mathrm{Ï€}}{2}\\ \sin \frac{\mathrm{Ï€}}{2}& \cos \frac{\mathrm{Ï€}}{2}\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$

Multiply the matrices.

$\left(\begin{array}{cc}\cos \frac{\mathrm{Ï€}}{2}& -\sin \frac{\mathrm{Ï€}}{2}\\ \sin \frac{\mathrm{Ï€}}{2}& \cos \frac{\mathrm{Ï€}}{2}\end{array}\right)\left(\begin{array}{cc}\cos \frac{\mathrm{Ï€}}{2}& -\sin \frac{\mathrm{Ï€}}{2}\\ \sin \frac{\mathrm{Ï€}}{2}& \cos \frac{\mathrm{Ï€}}{2}\end{array}\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$

Substitute $\mathrm{θ}=\mathrm{ϕ}=\mathrm{π}/2$into the right-hand side of the relation (6.14).

role="math" localid="1658989214239" $\left(\begin{array}{cc}\cos \left(\frac{\mathrm{Ï€}}{2}+\frac{\mathrm{Ï€}}{2}\right)& -\sin \left(\frac{\mathrm{Ï€}}{2}+\frac{\mathrm{Ï€}}{2}\right)\\ \sin \left(\frac{\mathrm{Ï€}}{2}+\frac{\mathrm{Ï€}}{2}\right)& \cos \left(\frac{\mathrm{Ï€}}{2}+\frac{\mathrm{Ï€}}{2}\right)\end{array}\right)=\left(\begin{array}{cc}\cos \left(\mathrm{Ï€}\right)& -\sin \left(\mathrm{Ï€}\right)\\ \sin \left(\mathrm{Ï€}\right)& \cos \left(\mathrm{Ï€}\right)\end{array}\right)$

Evaluate the sine and cosine function.

$\left(\begin{array}{cc}\cos \left(\mathrm{Ï€}\right)& -\sin \left(\mathrm{Ï€}\right)\\ \sin \left(\mathrm{Ï€}\right)& \cos \left(\mathrm{Ï€}\right)\end{array}\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$

Hence, the relation (6.14) is verified numerically