the following equation involves multiple angles. solve the equation on the interval 0, 2π). tan x/2 = √3/3…

the following equation involves multiple angles. solve the equation on the interval 0, 2π). tan x/2 = √3/3 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. x = (type an exact answer using π as needed. use integers or fractions for any numbers in the expression. use comma to separate answers as needed.) b. there is no solution.

the following equation involves multiple angles. solve the equation on the interval 0, 2π). tan x/2 = √3/3 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. x = (type an exact answer using π as needed. use integers or fractions for any numbers in the expression. use comma to separate answers as needed.) b. there is no solution.

Answer

Explanation:

Step1: Recall inverse - tangent property

If $\tan\theta = a$, then $\theta=\arctan(a)+k\pi$, where $k\in\mathbb{Z}$. Here $\theta = \frac{x}{2}$ and $a=\frac{\sqrt{3}}{3}$. So, $\frac{x}{2}=\arctan(\frac{\sqrt{3}}{3})+k\pi$. Since $\arctan(\frac{\sqrt{3}}{3})=\frac{\pi}{6}$, we have $\frac{x}{2}=\frac{\pi}{6}+k\pi$.

Step2: Solve for $x$

Multiply both sides of the equation $\frac{x}{2}=\frac{\pi}{6}+k\pi$ by 2 to get $x = \frac{\pi}{3}+2k\pi$.

Step3: Find solutions in the interval $[0,2\pi)$

When $k = 0$, $x=\frac{\pi}{3}$. When $k = 1$, $x=\frac{\pi}{3}+2\pi>\ 2\pi$.

Answer:

A. $x=\frac{\pi}{3}$