Chapter 5.4, Problem 23E

Solution

To find: The solution of the given equation in the interval $\left[0,2\pi \right)$ and use graph verify the solution.

The solutions are $x=\frac{\pi }{3},\pi ,\frac{5\pi }{3}$ .

Given data:

The equation is $\cos 2x+\cos x=0$ .

Formula used:

Use the double angle identity for cosine.

  $\cos 2u=2{\cos }^{2}u-1$

Calculation:

Simplify the given equation by using double angle identity for cosine.

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

Simplify, further.

  $\left(\cos x+1\right)\left(2\cos x-1\right)=0$

Use the zero factor property. Equate the first factor to zero.

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

Equate the second factor to zero.

  $\begin{array}{l}2\cos x-1=0\\ \cos x=\frac{1}{2}\\ x={\cos }^{-1}\left(\frac{1}{2}\right)\\ x=\frac{\pi }{3},\frac{5\pi }{3}\end{array}$

Use graphical calculator to draw the graph for the given equation.

  

From the above graph, it is clear that the solutions are $x=\frac{\pi }{3},\pi ,\frac{5\pi }{3}$ .

Therefore, the solutions are $x=\frac{\pi }{3},\pi ,\frac{5\pi }{3}$ .