Answer
$sin ~\theta = -\frac{1}{2}$
$cos ~\theta = -\frac{\sqrt{3}}{2}$
$tan ~\theta = \frac{\sqrt{3}}{3}$
$csc ~\theta = -2$
$sec ~\theta = -\frac{2\sqrt{3}}{3}$
$cot ~\theta = \sqrt{3}$
Work Step by Step
$-150^{\circ}+360^{\circ} = 210^{\circ} = 180^{\circ}+30^{\circ}$
The angle $-150^{\circ}$ makes an angle of $30^{\circ}$ below the negative x-axis. We can let $x=-\sqrt{3}$ and $y=-1$. Then $r = 2$.
We can find the values of the six trigonometric functions:
$sin ~\theta = \frac{y}{r} = \frac{-1}{2}$
$cos ~\theta = \frac{x}{r} = \frac{-\sqrt{3}}{2}$
$tan ~\theta = \frac{y}{x} = \frac{-1}{-\sqrt{3}} = \frac{\sqrt{3}}{3}$
$csc ~\theta = \frac{r}{y} = \frac{2}{-1} = -2$
$sec ~\theta = \frac{r}{x} = \frac{2}{-\sqrt{3}} = -\frac{2\sqrt{3}}{3}$
$cot ~\theta = \frac{x}{y} = \frac{-\sqrt{3}}{-1} = \sqrt{3}$
