Answer
positive; III; $30^{\circ}$; $\frac{\sqrt 3}{3}$
Work Step by Step
We need to find the angle coterminal with $-150^{\circ}$ and is in between $0^{\circ}$ and $360^{\circ}$.
$-150^{\circ}$ is coterminal with $-150^{\circ}+360^{\circ}=210^{\circ}$.
$210^{\circ}$ is in the third quadrant.
In the third quadrant, the value of $\tan\theta$ is positive.
For the third quadrant, reference angle
$\theta'=\theta-180^{\circ}=210^{\circ}-180^{\circ}=30^{\circ}$
$\tan -150^{\circ}=\tan 210^{\circ}=+\tan\theta'=\tan 30^{\circ}=\frac{1}{\sqrt 3}=\frac{\sqrt 3}{3}$
