Chapter 7.5, Problem 49E

To determine

To find: The solution of equation $\cos 2\theta -\cos 4\theta$ in the interval $\left[0,2\pi \right]$ .

Answer to Problem 49E

The solution of the equation $\cos 2\theta -\cos 4\theta$ in interval are $0,\pi ,\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3}$ and $\frac{5\pi }{3}$ .

Explanation of Solution

Given information:

The equation is $\cos 2\theta -\cos 4\theta$ .

Calculation:

Write the double angle formula.

  $\cos 4\theta =2{\cos }^2\theta -1$

Simplify the equation using double angle formula

  $\begin{array}{c}\cos 2\theta -\left(2{\cos }^2\theta -1\right)=0\\ 2{\cos }^2\theta -\cos 2\theta -1=0\\ \left(2\cos 2\theta +1\right)\left(\cos 2\theta -1\right)=0\end{array}$

Further solve for $2\cos 2\theta +1=0$

  $\begin{array}{c}2\cos 2\theta +1=0\\ \cos 2\theta =\frac{-1}{2}\\ \theta =n\pi \text{ +}\frac{\pi }{3}\text{or}n\pi \text{ +}\frac{2\pi }{3}\end{array}$

In above equation $n$ is integer.

Further solve the above value for given range.

  $\theta =\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3},\frac{5\pi }{3}$

Solve $\left(\cos 2\theta -1\right)=0$ .

  $\begin{array}{c}\cos 2\theta -1=0\\ 2\theta =0\\ \theta =0\text{or}\pi \end{array}$

Therefore, the solution of the equation $\cos 2\theta -\cos 4\theta$ in interval are $0,\pi ,\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3}$ and $\frac{5\pi }{3}$ .