Question

Do problem 26if $\mathbf{\theta }\mathbf{=}\mathbf{\pi }\mathbf{/}\mathbf{2}\mathbf{,}\mathbf{\varphi }\mathbf{=}\mathbf{\pi }\mathbf{/}\mathbf{4}$ .

Short answer

The relation $\left(\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right)\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)=\left(\begin{array}{cc}\cos \left(\theta +\varphi \right)& -\sin \left(\theta +\varphi \right)\\ \sin \left(\theta +\varphi \right)& \cos \left(\theta +\varphi \right)\end{array}\right)$ is verified numerically

for $\theta =\pi /2$and $\varphi =\pi /4$, its numerical value is $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)$.

Step-by-step solution

The trigonometric values of sine and cosine function for different angles:

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

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

Given parameters:

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

It needs to be verified for $\theta =\pi /2$and role="math" localid="1658982863554" $\mathrm{\varphi }=\mathrm{\pi }/4$.

Finding the numerical values of the left-hand side and the right-hand side of the relation:

Substitute $\theta =\pi /2$ and $\varphi =\pi /4$into the left-hand side of the relation and evaluate the result.

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

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 $\theta =\pi /2$and $\varphi =\pi /4$into the right-hand side of the relation (6.14).

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

Evaluate the sine and cosine function.

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

Hence, the relation (6.14) is verified.