Question

which of the following is true of the location of an angle, $\theta$, whose tangent value is $-\frac{sqrt{3}}{3}$?
$\theta$ has a 30 - degree reference angle and is located in quadrant ii or iv
$\theta$ has a 30 - degree reference angle and is located in quadrant ii or iii
$\theta$ has a 60 - degree reference angle and is located in quadrant ii or iv
$\theta$ has a 60 - degree reference angle and is located in quadrant ii or iii

Explanation:

Step1: Recall tangent - reference angle relationship

We know that $\tan\theta=\frac{\sin\theta}{\cos\theta}$. The reference - angle $\alpha$ is related to the angle $\theta$ in the standard position. We know that $\tan30^{\circ}=\frac{\sqrt{3}}{3}$ and $\tan60^{\circ}=\sqrt{3}$. Since $\tan\theta =-\frac{\sqrt{3}}{3}$, we consider the reference - angle whose tangent has an absolute value of $\frac{\sqrt{3}}{3}$, which is $30^{\circ}$ or $\frac{\pi}{6}$ radians.

Step2: Determine the quadrants

The tangent function is negative in Quadrant II ($\sin\theta>0,\cos\theta < 0$) and Quadrant IV ($\sin\theta<0,\cos\theta>0$).

Answer:

A. $\theta$ has a 30 - degree reference angle and is located in Quadrant II or IV