Question

solve, finding all solutions. tan x = -√3. x = □ + πk, where k is any integer. (type an exact answer, using π as needed. type any angle in radians between 0 and π.)

Explanation:

Step1: Recall tangent values

We know that $\tan x = -\sqrt{3}$. The tangent function has period $\pi$. The reference - angle for which $\tan\theta=\sqrt{3}$ is $\frac{\pi}{3}$. Since $\tan x<0$, $x$ is in the second or fourth quadrant.

Step2: Find the principal solutions

In the second quadrant, $x=\pi - \frac{\pi}{3}=\frac{2\pi}{3}$, and in the fourth quadrant, $x = 2\pi-\frac{\pi}{3}=\frac{5\pi}{3}$. The general solution of $\tan x = -\sqrt{3}$ is $x=\frac{2\pi}{3}+\pi k$, where $k$ is an integer.

Answer:

$\frac{2\pi}{3}$