Chapter 7.5, Problem 55E

To determine

To find: The solution of the equation $\cos 4\theta +\cos 2\theta =\cos \theta$ using a Sum-to-Product formula.

Answer to Problem 55E

The solution of the equation $\cos 4\theta +\cos 2\theta =\cos \theta$ is

  $\theta =2n\pi +\frac{\pi }{2}\text{or}\theta =2n\pi +\frac{3\pi }{2}\text{or}\theta =\frac{2}{3}k\pi +\frac{\pi }{9}\text{or}\theta =\frac{2}{3}k\pi +\frac{5\pi }{9}$ .

Explanation of Solution

Given information:

The equation $\cos 4\theta +\cos 2\theta =\cos \theta$

Calculation:

Write the sum-to-product formula.

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

Simplify the given equation using the above formula.

  $\begin{array}{c}\cos 4\theta +\cos 2\theta =\cos \theta \\ 2\cos \frac{4\theta +2\theta }{2}\cos \frac{4\theta -2\theta }{2}=\cos \theta \\ 2\cos 3\theta \cos \theta =\cos \theta \\ 2\cos 3\theta \cos \theta -\cos \theta =0\end{array}$

  $\begin{array}{c}\cos \theta \left(2\cos 3\theta -1\right)=0\\ \cos \theta =0\text{or}\cos 3\theta =\frac{1}{2}\end{array}$

Solve $\cos \theta =0$ .

  $\begin{array}{l}\theta =\frac{\pi }{2},\frac{3\pi }{2}\\ \theta =2n\pi +\frac{\pi }{2}\text{or}\theta =2n\pi +\frac{3\pi }{2}\end{array}$

Solve $\cos 3\theta =\frac{1}{2}$ .

  $\begin{array}{l}3\theta =\frac{\pi }{3},\frac{5\pi }{3}\\ 3\theta =2nP+\frac{\pi }{3}\text{or}3\theta =2n\pi +\frac{5\pi }{3}\\ \theta =\frac{2}{3}n\pi +\frac{\pi }{9}\text{or}\theta =\frac{2}{3}n\pi +\frac{5\pi }{9}\end{array}$

Here $n$ is any integer.

Therefore, the solution of the equation $\cos 4\theta +\cos 2\theta =\cos \theta$ is

  $\theta =2n\pi +\frac{\pi }{2}\text{or}\theta =2n\pi +\frac{3\pi }{2}\text{or}\theta =\frac{2}{3}k\pi +\frac{\pi }{9}\text{or}\theta =\frac{2}{3}k\pi +\frac{5\pi }{9}$ .