Chapter 14.8, Problem 16WE

To determine

To find:the solution of the given trigonometric function.

Answer to Problem 16WE

  $\theta ={30}^{\circ },{150}^{\circ }$

Explanation of Solution

Given:

The given trigonometric function is $2\cos 2\theta =1$ .

Calculation:

As thegiven trigonometric function is $2\cos 2\theta =1$ .

Now, to solve the given trigonometric function. Therefore,

  $\begin{array}{l}2\cos 2\theta =1\{\text{given equation}}\\ \cos 2\theta =\frac{1}{2}\left\{\text{divide both sides by }2\right\}\\ \therefore 2\theta ={\cos }^{-1}\frac{1}{2}\{\text{inverse trigonometry property}}\\ 2\theta ={\cos }^{-1}\left(\cos {60}^{\circ }\right)\{\left(\cos {60}^{\circ }\right)=\frac{1}{2}\text{}}\\ \Rightarrow 2\theta ={\cos }^{-1}\left(\cos {300}^{\circ }\right)\{\text{also,}\left(\cos {150}^{\circ }\right)=\frac{1}{2}\text{ in quad IV}}\\ \therefore 2\theta ={60}^{\circ }\text{ and }2\theta ={300}^{\circ }\\ \therefore \theta ={30}^{\circ },{150}^{\circ }\end{array}$

Hence, the solution of the given trigonometric function is $\theta ={30}^{\circ },{150}^{\circ }$ .