Answer
1. $sin\theta= \frac{3}{5}$,
2. $cos\theta= -\frac{4}{5}$,
3. $tan\theta= -\frac{3}{4}$,
4. $cot\theta= -\frac{4}{3}$,
5. $csc\theta= \frac{5}{3}$.
Work Step by Step
Given $sec\theta=-\frac{5}{4}\gt0$ and $tan\theta\lt0$, we know $\theta$ is in quadrant II, let $r=5, x=-4$, we have $y=\sqrt {(5)^2-(-4)^2}=3$, thus:
1. $sin\theta=\frac{y}{r}=\frac{3}{5}$,
2. $cos\theta=\frac{x}{r}=-\frac{4}{5}$,
3. $tan\theta=\frac{y}{x}=-\frac{3}{4}$,
4. $cot\theta=\frac{1}{tan\theta}=-\frac{4}{3}$,
5. $csc\theta=\frac{1}{sin\theta}=\frac{5}{3}$.
