Chapter 7.4, Problem 47E

To determine

To find:The solution of the equation ${\cos }^2\theta -\cos \theta -6=0$ .

Answer to Problem 47E

There is no solution possible solution for the equation ${\cos }^2\theta -\cos \theta -6=0$ .

Explanation of Solution

Given information:

The given equation is ${\cos }^2\theta -\cos \theta -6=0$ .

Calculation:

Simplify the given equation.

  $\begin{array}{c}{\cos }^2\theta -\cos \theta -6=0\\ {\cos }^2\theta -3\cos \theta +2\cos \theta -6=0\\ \left(\cos \theta +2\right)\left(\cos \theta -3\right)=0\end{array}$

First substitute the factor $\cos \theta +2=0$ .

Simplify the equation

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

As the value of the function cosine is always between $-1$ to $1$ . Therefore, $\cos \theta +2=0$ is not possible for any value of $\theta$ .

Now substitute the factor $\cos \theta -3=0$ .

As the value of the function cosine is always between $-1$ to $1$ . Therefore, $\cos \theta -3=0$ is not possible for any value of $\theta$ .

Therefore, there is no solution possible solution for the equation ${\cos }^2\theta -\cos \theta -6=0$ .