Answer
1. $cos\theta=-\frac{\sqrt {5}}{3}$
2. $tan\theta=\frac{2\sqrt {5}}{2}$
3. $cot\theta= \frac{\sqrt {5}}{2}$
4. $sec\theta= -\frac{3\sqrt {5}}{5}$
5. $csc\theta= -\frac{3}{2}$
Work Step by Step
Given $sin\theta=-\frac{2}{3}$ and $\theta$ in quadrant III, we have $y=-2, r=3, x=-\sqrt {3^2-(-2)^2}=-\sqrt {5}$, thus
1. $cos\theta=-\frac{\sqrt {5}}{3}$
2. $tan\theta=\frac{2\sqrt {5}}{2}$
3. $cot\theta=\frac{1}{tan\theta}=\frac{\sqrt {5}}{2}$
4. $sec\theta=\frac{1}{cos\theta}=-\frac{3\sqrt {5}}{5}$
5. $csc\theta=\frac{1}{sin\theta}=-\frac{3}{2}$
