the following equation involves multiple angl...
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. $x=\frac{\pi}{3}$