Chapter 7.4, Problem 30E

To determine

To find: The solution of the equation $4\cos \theta +1=0$ .

Answer to Problem 30E

The solutions of the equation $4\cos \theta +1=0$ are $\theta =1.823+2k\pi ,4.46+2k\pi$ .

Explanation of Solution

Given information:

The given equation is $4\cos \theta +1=0$ .

Calculation:

Simplify the given equation.

  $\begin{array}{c}4\cos \theta +1=0\\ \cos \theta =\frac{-1}{4}\\ =-0.25\end{array}$

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

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

  $\begin{array}{c}\theta ={\cos }^{-1}\left(-0.25\right)\\ \approx 1.823,4.46\end{array}$

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

So, The general solution of the equation is given by

  $\theta =1.823+2k\pi ,4.46+2k\pi$ .

Therefore, the solutions of the equation $4\cos \theta +1=0$ are $\theta =1.823+2k\pi ,4.46+2k\pi$ .