Question

Verify (6.14) by multiplying the matrices and using trigonometric addition formulas.

Short answer

The result $\left(\begin{array}{cc}\cos \mathrm{f}& -\sin \mathrm{f}\\ \sin \mathrm{f}& \cos \mathrm{f}\end{array}\right)\left(\begin{array}{cc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }\\ \sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right)=\left(\begin{array}{cc}\cos (\mathrm{\theta }+\mathrm{f})& -\sin (\mathrm{\theta }+\mathrm{f})\\ \sin (\mathrm{\theta }+\mathrm{f})& \cos (\mathrm{\theta }+\mathrm{f})\end{array}\right)$is verified by using the trigonometric addition formulas.

Step-by-step solution

The trigonometric addition formulas:

The trigonometric addition formulas for the cosine and sine functions.

$$\begin{gathered} \mathbf{cos}\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}\mathbf{=}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{cos}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{-}\mathbf{sin}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{sin}\mathbf{\left(}\mathbf{b}\mathbf{\right)} \\ \mathbf{sin}\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}\mathbf{=}\mathbf{sin}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{cos}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{+}\mathbf{cos}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{sin}\mathbf{\left(}\mathbf{b}\mathbf{\right)} \end{gathered}$$

Given parameters

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

It needs to be verified.

Multiply the matrices on the left-hand side

Multiply the matrices

$\left(\begin{array}{cc}\cos \mathrm{f}& -\sin \mathrm{f}\\ \sin \mathrm{f}& \cos \mathrm{f}\end{array}\right)\left(\begin{array}{cc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }\\ \sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right)=\left(\begin{array}{cc}\cos f.\cos \theta -\sin f.\sin \theta & -\sin f.\mathrm{co}s\theta +\sin f.\cos \theta \\ \sin f.\cos \theta +\sin f.\cos \theta & \cos f.\cos \theta -\sin f.\sin \theta \end{array}\right)$

Use the trigonometric addition formulas for the cosine and sine function.

$\left(\begin{array}{cc}\cos \mathrm{f}& -\sin \mathrm{f}\\ \sin \mathrm{f}& \cos \mathrm{f}\end{array}\right)\left(\begin{array}{cc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }\\ \sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right)=\left(\begin{array}{cc}\cos (\mathrm{\theta }+\mathrm{f})& -\sin (\mathrm{\theta }+\mathrm{f})\\ \sin (\mathrm{\theta }+\mathrm{f})& \cos (\mathrm{\theta }+\mathrm{f})\end{array}\right)$

Hence, (6.14) is verified.