Chapter 7.4, Problem 22E

To determine

To find: The solution of the equation $\tan \theta =2.5$ and list six specific solutions.

Answer to Problem 22E

The solutions of the equation $\tan \theta =2.5$ are $\theta =1.19+k\pi$ and six specific solutions are $-5.09,-1.95,1.19,4.33,7.47,10.61$ .

Explanation of Solution

Given information:

The given equation is $\tan \theta =2.5$ .

Calculation:

As tangent has period $\pi$ , So first find the solutions in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ .

As the given equation is $\tan \theta =2.5$ .

Therefore, the solution to the equation in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ are

  $\begin{array}{c}\theta ={\tan }^{-1}\left(2.5\right)\\ \approx 1.19\end{array}$

As tangent repeats its values after every $\pi$ units.

So, The general solution of the equation is given by

  $\theta =1.19+k\pi$ .

Now substitute $k=-2,-1,0,1,2,3$ in the above solution

  $\begin{array}{c}\theta =1.19-2\pi ,1.19-\pi ,1.19,1.19+\pi ,1.19+2\pi ,1.19+3\pi \\ =-5.09,-1.95,1.19,4.33,7.47,10.61\end{array}$

Therefore the six specific solutions of the equation $\tan \theta =2.5$ are $-5.09,-1.95,1.19,4.33,7.47,10.61$ .

Therefore, the solutions of the equation $\tan \theta =2.5$ are $\theta =1.19+k\pi$ .