91Ó°ÊÓ

The numerical values of the sine and cosine function at $\frac{\mathbf{Ï€}}{\mathbf{2}}$, $\frac{\mathbf{Ï€}}{\mathbf{4}}$and $\frac{\mathbf{3}\mathbf{Ï€}}{\mathbf{4}}$ are,

$$\begin{gathered} \mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\frac{\mathbf{Ï€}}{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{1} \\ \mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\frac{\mathbf{Ï€}}{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{0} \\ \mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\frac{\mathbf{Ï€}}{\mathbf{4}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}}} \\ \mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\frac{\mathbf{Ï€}}{\mathbf{4}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}}} \\ \mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\frac{\mathbf{3}\mathbf{Ï€}}{\mathbf{4}}\mathbf{\right)}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\sqrt{\mathbf{2}}} \end{gathered}$$

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

It needs to be verified for $\mathrm{θ}=\mathrm{π}/2$and role="math" localid="1658982863554" $\mathrm{ϕ}=\mathrm{π}/4$.

Substitute $\mathrm{θ}=\mathrm{π}/2$ and $\mathrm{ϕ}=\mathrm{π}/4$into the left-hand side of the relation and evaluate the result.

$\left(\begin{array}{cc}\cos \frac{\mathrm{Ï€}}{4}& -\sin \frac{\mathrm{Ï€}}{4}\\ \sin \frac{\mathrm{Ï€}}{4}& \cos \frac{\mathrm{Ï€}}{4}\end{array}\right)\left(\begin{array}{cc}\cos \frac{\mathrm{Ï€}}{2}& -\sin \frac{\mathrm{Ï€}}{2}\\ \sin \frac{\mathrm{Ï€}}{2}& \cos \frac{\mathrm{Ï€}}{2}\end{array}\right)=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$

Multiply the matrices.

$\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)$

Substitute $\mathrm{θ}=\mathrm{π}/2$and $\mathrm{ϕ}=\mathrm{π}/4$into the right-hand side of the relation (6.14).

$\left(\begin{array}{cc}\cos \left(\begin{array}{cc}\frac{\mathrm{Ï€}}{2}+& \frac{\mathrm{Ï€}}{4}\end{array}\right)& -\sin \left(\begin{array}{cc}\frac{\mathrm{Ï€}}{2}+& \frac{\mathrm{Ï€}}{4}\end{array}\right)\\ \sin \left(\begin{array}{cc}\frac{\mathrm{Ï€}}{2}+& \frac{\mathrm{Ï€}}{4}\end{array}\right)& \cos \left(\begin{array}{cc}\frac{\mathrm{Ï€}}{2}+& \frac{\mathrm{Ï€}}{4}\end{array}\right)\end{array}\right)=\left(\begin{array}{cc}\cos \left(\frac{3\mathrm{Ï€}}{4}\right)& -\sin \left(\frac{3\mathrm{Ï€}}{4}\right)\\ \sin \left(\frac{3\mathrm{Ï€}}{4}\right)& \cos \left(\frac{3\mathrm{Ï€}}{4}\right)\end{array}\right)$

Evaluate the sine and cosine function.

$\left(\begin{array}{cc}\cos \left(\frac{3\mathrm{Ï€}}{4}\right)& -\sin \left(\frac{3\mathrm{Ï€}}{4}\right)\\ \sin \left(\frac{3\mathrm{Ï€}}{4}\right)& \cos \left(\frac{3\mathrm{Ï€}}{4}\right)\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)$

Hence, the relation (6.14) is verified.