Chapter 3: Q25P (page 123)

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

Short Answer

Expert verified

The result $(\begin{matrix} \cos f & - \sin f \\ \sin f & \cos f \end{matrix}) (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) = (\begin{matrix} \cos (θ + f) & - \sin (θ + f) \\ \sin (θ + f) & \cos (θ + f) \end{matrix})$ 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.

$\cos (a + b) = \cos (a) \cos (b) - \sin (a) \sin (b) \sin (a + b) = \sin (a) \cos (b) + \cos (a) \sin (b)$

Given parameters

The result (6.14) is $(\begin{matrix} \cos f & - \sin f \\ \sin f & \cos f \end{matrix}) (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) = (\begin{matrix} \cos (θ + f) & - \sin (θ + f) \\ \sin (θ + f) & \cos (θ + f) \end{matrix})$ .

It needs to be verified.

Multiply the matrices on the left-hand side

Multiply the matrices

$(\begin{matrix} \cos f & - \sin f \\ \sin f & \cos f \end{matrix}) (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) = (\begin{matrix} \cos f . \cos θ - \sin f . \sin θ & - \sin f . co s θ + \sin f . \cos θ \\ \sin f . \cos θ + \sin f . \cos θ & \cos f . \cos θ - \sin f . \sin θ \end{matrix})$

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

$(\begin{matrix} \cos f & - \sin f \\ \sin f & \cos f \end{matrix}) (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) = (\begin{matrix} \cos (θ + f) & - \sin (θ + f) \\ \sin (θ + f) & \cos (θ + f) \end{matrix})$

Hence, (6.14) is verified.

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

Short Answer

Step by step solution

The trigonometric addition formulas:

Given parameters

Multiply the matrices on the left-hand side

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