Chapter 13, Problem 22PT

To determine

To find : The value of $\theta$ .

Answer to Problem 22PT

  $\theta =\frac{2\pi }{3}$

Explanation of Solution

Given information :

We know that, the given expression is $2{\cos }^2\theta -3\cos \theta -2=0$

Calculation :

The given equation is

  $\begin{array}{l}2{\cos }^2\theta -3\cos \theta -2=0\\ \Rightarrow 2{\cos }^2\theta -4\cos \theta +\cos \theta -2=0\\ \Rightarrow 2\cos \theta [\cos \theta -2]+1[\cos \theta -2]=0\\ \Rightarrow (2\cos \theta +1)(\cos \theta -2)=0\\ \Rightarrow 2\cos \theta +1=0or\cos \theta -2=0\\ \Rightarrow \cos \theta =-\frac{1}{2}or\cos \theta =2\\ but\cos \theta =2,\text{ not possible as}\\ \cos \theta \in [-1,1]\\ \Rightarrow \cos \theta =-\frac{1}{2}=\cos \left(\pi -\frac{\pi }{3}\right)\\ \Rightarrow \theta =\frac{2\pi }{3}\end{array}$