Question

find two angles between 0 and 2π for the given condition. express your answer in terms of radians. do not use decimals in your answer. tanθ = -√3/3 the angles are and .

Explanation:

Step1: Recall tangent - angle relationship

We know that $\tan\theta=\frac{\sin\theta}{\cos\theta}$, and the tangent function has a period of $\pi$. The principal - value of $\theta$ for which $\tan\theta =-\sqrt{3}$ can be found from the unit - circle or the special right - triangles. We know that $\tan\frac{\pi}{3}=\sqrt{3}$.

Step2: Determine the angles in the given range

Since $\tan\theta$ is negative, $\theta$ lies in the second or fourth quadrant.
In the second quadrant, $\theta=\pi-\frac{\pi}{3}=\frac{2\pi}{3}$.
In the fourth quadrant, $\theta = 2\pi-\frac{\pi}{3}=\frac{5\pi}{3}$.

Answer:

$\frac{2\pi}{3}$, $\frac{5\pi}{3}$