Chapter 14.8, Problem 42WE

To determine

To find:the solution of the given trigonometric function.

Answer to Problem 42WE

  $x=\frac{7\pi }{24},\frac{19\pi }{24}$

Explanation of Solution

Given:

The given trigonometric function is $\tan \left(2x+\frac{\pi }{3}\right)=1$ .

Calculation:

As thegiven trigonometric function is $\tan \left(2x+\frac{\pi }{3}\right)=1$ .

Now, to solve the given trigonometric function. Therefore,

  $\begin{array}{l}\tan \left(2x+\frac{\pi }{3}\right)=1\{\text{given equation}}\\ \therefore \left(2x+\frac{\pi }{3}\right)={\tan }^{-1}1\\ \Rightarrow \left(2x+\frac{\pi }{3}\right)={\tan }^{-1}\left(\tan \frac{\pi }{4}\right)\{tan\frac{\pi }{4}=1\}\\ also,\left(2x+\frac{\pi }{3}\right)={\tan }^{-1}\left(\tan \frac{5\pi }{4}\right)\{tan\frac{5\pi }{4}=1\}\\ \therefore 2x+\frac{\pi }{3}=\frac{\pi }{4};2x+\frac{\pi }{3}=\frac{5\pi }{4}\\ \Rightarrow x=\frac{7\pi }{24};x=\frac{19\pi }{24}\end{array}$

Hence, the solution of the given trigonometric function is $x=\frac{7\pi }{24},\frac{19\pi }{24}$ .