Chapter 7.4, Problem 9E

To determine

To find: The solution of the equation $\cos \theta =\frac{1}{4}$ .

Answer to Problem 9E

The solutions of the equation $\cos \theta =\frac{1}{4}$ are $\theta =1.32+2k\pi$ and $\theta =4.97+2k\pi$ .

Explanation of Solution

Given information:

The given equation is $\cos \theta =\frac{1}{4}$ .

Calculation:

As cosine has period $2\pi$ , So first find the solutions in $\left[0,2\pi \right)$ . The function cosine is positive in the first and the fourth quadrants.

Therefore, the solution to the equation in $\left[0,2\pi \right)$ are

  $\begin{array}{c}\theta ={\cos }^{-1}\left(\frac{1}{4}\right)\\ \approx 1.32,4.97\end{array}$

As cosine repeats its values after every $2\pi$ units.

So, the general solution of the equation is given by

  $\theta =1.32+2k\pi ,4.97+2k\pi$

Therefore, the solutions of the equation $\cos \theta =\frac{1}{4}$ are $\theta =1.32+2k\pi$ and $\theta =4.97+2k\pi$ .