Chapter 7.5, Problem 45E

To determine

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

Answer to Problem 45E

The solution of equation $\cos 2\theta +\cos \theta =2$ in the interval $\left[0,2\pi \right]$ is $0$ .

Explanation of Solution

Given information:

The equation is $\cos 2\theta +\cos \theta =2$ .

Calculation:

Write the double angle formula.

  $\cos 2\theta =2{\cos }^2\theta -1$

Simplify the equation using double angle formula

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

Further solve $2\cos \theta +3$ .

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

The value of cosine can never be less than $-1$ . Therefore, the above value will be neglected.

Solve for $\cos \theta -1=0$ .

  $\begin{array}{c}\cos \theta -1=0\\ \cos \theta =1\\ \theta =0\end{array}$

Therefore, the solution of equation $\cos 2\theta +\cos \theta =2$ in the interval $\left[0,2\pi \right]$ is $0$ .