Chapter 7.7, Problem 44AYU

solve each equation on the interval $0\le \theta \le \text{ 2}\pi$

$\cos \left(2\theta \right)\text{ + cos}\left(4\theta \right)\text{ = 0 }$

To determine

To solve: $\cos 2\theta +\cos 4\theta$ for $0\le \theta <2\pi$

Answer to Problem 44AYU

Solution:

The solution of $\cos 2\theta +\cos 4\theta =\frac{\pi }{2},\frac{3\pi }{2}$.

Explanation of Solution

Given:

$\cos 2\theta +\cos 4\theta$, $0\le \theta <2\pi$

Calculation:

By Sum of product formula,

$\cos ά+\cos \beta =2\cos \frac{ά+\beta }{2}\cos \frac{ά-\beta }{2}$

$2\cos \frac{2\theta +4\theta }{2}\cos \frac{2\theta -4\theta }{2}$

$2\cos 3\theta \cos \left(-\theta \right)$

$\text{cosine}$ is an even function.

$2\cos 3\theta \cos \theta$

To solve,

$\cos 3\theta =0,\text{ cos}\theta =0$

$\theta =\frac{\pi }{2},\frac{3\pi }{2}$