Chapter 7.5, Problem 54E

To determine

To find: The solution of the equation $\cos 5\theta -\cos 7\theta =0$ using a Sum-to-Product formula.

Answer to Problem 54E

The solution of the equation $\cos 5\theta -\cos 7\theta =0$ is $\theta =\frac{n\pi }{6}\text{or }\theta =n\pi$ .

Explanation of Solution

Given information:

The equation $\cos 5\theta -\cos 7\theta =0$ .

Calculation:

Write the sum-to-product formula.

  $\cos c-\cos d=-2\sin \frac{c+d}{2}\sin \frac{c-d}{2}$

Simplify the given equation using the above formula.

  $\begin{array}{c}\cos 5\theta -\cos 7\theta =0\\ -2\sin \frac{5\theta +7\theta }{2}\sin \frac{5\theta -7\theta }{2}=0\\ -2\sin 6\theta \sin \left(-\theta \right)=0\\ \sin 6\theta =0\text{or sin}\theta \text{=0}\end{array}$

Solve for $\sin 6\theta =0$ .

  $\begin{array}{l}6\theta =n\pi \\ \theta =\frac{n\pi }{6}\end{array}$

Solve for $\sin \theta =0$ .

  $\theta =n\pi$

Here $n$ is any integer.

The general solution of $\theta$ is expressed as,

  $\theta =\frac{n\pi }{6}\text{or }\theta =n\pi$

Therefore, the solution of the equation $\cos 5\theta -\cos 7\theta =0$ is $\theta =\frac{n\pi }{6}\text{or }\theta =n\pi$ .