Chapter 5, Problem 51CR

To determine

To evaluate: the solution of given trigonometric solution.

Answer to Problem 51CR

The solution of given equation is

  $x=2\pi n,x=\frac{2\pi }{3}+2\pi n,\frac{4\pi }{3}+2\pi n$

Explanation of Solution

Given:

  $2{\cos }^{2}x-\cos x=1$

It is given that

  $2{\cos }^{2}x-\cos x=1$

To rewrite the expression, let $\cos x=y$

  $\begin{array}{l}2{\cos }^{2}x-\cos x=1\\ 2y^2-y-1=0\end{array}$

Applying the quadratic equation,

  $\begin{array}{l}y=\frac{1\pm \sqrt{1-4\left(2\right)\left(-1\right)}}{4}\\ y=\frac{1\pm \sqrt{1+8}}{4}\\ y=\frac{1\pm 3}{4}\\ y=1,-\frac{1}{2}\end{array}$

So,

  $\begin{array}{l}\cos x=1\\ x=2\pi n\end{array}$

Further,

  $\begin{array}{l}\cos x=-\frac{1}{2}\\ x=\frac{2\pi }{3}+2\pi n,\frac{4\pi }{3}+2\pi n\end{array}$

Hence, the solution of given equation is

  $x=2\pi n,x=\frac{2\pi }{3}+2\pi n,\frac{4\pi }{3}+2\pi n$

.