find all solutions of the equation in the interval 0,2π). tan x/3 = 1 select the correct choice below and…

find all solutions of the equation in the interval 0,2π). tan x/3 = 1 select the correct choice below and, if necessary, fill in the answer box. a. the solution set is (simplify your answer. type an exact answer, using π as needed. b. there is no solution.

find all solutions of the equation in the interval 0,2π). tan x/3 = 1 select the correct choice below and, if necessary, fill in the answer box. a. the solution set is (simplify your answer. type an exact answer, using π as needed. b. there is no solution.

Answer

Explanation:

Step1: Recall the inverse - tangent property

We know that if $\tan\theta = 1$, then $\theta=\frac{\pi}{4}+k\pi$, where $k\in\mathbb{Z}$. In our case, $\theta = \frac{x}{3}$, so $\frac{x}{3}=\frac{\pi}{4}+k\pi$.

Step2: Solve for x

Multiply both sides of the equation $\frac{x}{3}=\frac{\pi}{4}+k\pi$ by 3 to get $x = \frac{3\pi}{4}+3k\pi$.

Step3: Find solutions in the given interval

We want to find solutions in the interval $[0,2\pi)$. When $k = 0$, $x=\frac{3\pi}{4}$. When $k = 1$, $x=\frac{3\pi}{4}+3\pi=\frac{3\pi + 12\pi}{4}=\frac{15\pi}{4}>2\pi$.

Answer:

A. The solution set is $\left{\frac{3\pi}{4}\right}$