Chapter 7, Problem 136RE

To determine

To find:the exact solution of the given equation by using a value of $\cos \theta$ . Then find the form of the double-angle formula for $\cos \left(2\theta \right)$ and check which is most efficient to use.

Answer to Problem 136RE

  $\cos 2\theta =2{\cos }^2\theta -1$ .

Explanation of Solution

Given:

The given equation is $\cos \left(2\theta \right)$ .

Calculation:

According to the question, the most efficient formula to compute the value of $\cos \left(2\theta \right)$ .

To compute the value of $\cos \left(2\theta \right)$ , there are 3 formulas:

  $\begin{array}{l}\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta \\ \cos 2\theta =2{\cos }^2\theta -1\\ \cos 2\theta =1-2{\sin }^2\theta \end{array}$

Since, we are the given the value of $\cos \theta$ , the most efficient formula is one which contains only the value of $\cos \theta$ .

Hence, the most efficient formula is $\cos 2\theta =2{\cos }^2\theta -1$ .