Chapter 7.4, Problem 21E

To determine

To find: The solution of the equation $\cos \theta =0.28$ and list six specific solutions.

Answer to Problem 21E

The solutions of the equation $\cos \theta =0.28$ are $\theta =1.287+2k\pi ,4.996+2k\pi$ and six specific solutions are $-5,-1.29,1.29,5,7.57,11.28$ .

Explanation of Solution

Given information:

The given equation is $\cos \theta =0.28$ .

Calculation:

As cosine has period $2\pi$ , So first find the solutions in $\left[0,2\pi \right)$ .

As the given equation is $\cos \theta =0.28$ .

Function cosine is positive in first and fourth Quadrants.

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

  $\begin{array}{c}\theta ={\cos }^{-1}\left(0.28\right)\\ \approx 1.29,5.00\end{array}$

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

So, The general solution of the equation is given by

  $\theta =1.287+2k\pi ,4.996+2k\pi$ .

Now substitute $k=-1,0,1$ in the above solutions

  $\begin{array}{c}\theta =1.29-2\pi ,5-2\pi ,1.29,5,1.29+2\pi ,5+2\pi \\ =-5,-1.29,1.29,5,7.57,11.28\end{array}$

Therefore the six specific solutions of the equation $\cos \theta =0.28$ are $-5,-1.29,1.29,5,7.57,11.28$ .

Therefore, the solutions of the equation $\cos \theta =0.28$ are $\theta =1.287+2k\pi ,4.996+2k\pi$ .