Answer
$-\frac{1}{2}$, $-\frac{\sqrt 3}{2}$, $\frac{\sqrt 3}{3}$, $\sqrt 3$, $-\frac{2\sqrt 3}{3}$, and $-2$.
Work Step by Step
Given $\theta=-510^\circ=-360^\circ-150^\circ$ (quadrant III, reference angle $30^\circ$), we have $sin\theta =-sin30^\circ=-\frac{1}{2}$, $cos\theta =-cos30^\circ=-\frac{\sqrt 3}{2}$, $tan\theta =tan30^\circ=\frac{\sqrt 3}{3}$, $cot\theta = \frac{1}{tan\theta}=\sqrt 3$, $sec\theta = \frac{1}{cos\theta}=-\frac{2\sqrt 3}{3}$, and $csc\theta = \frac{1}{sin\theta}=-2$.
