Answer
1. $sin\theta =-\frac{12}{13}$,
2. $cos\theta =-\frac{5}{13}$,
3. $cot\theta =\frac{5}{12}$,
4. $sec\theta =-\frac{13}{5}$,
5. $csc\theta= -\frac{13}{12}$.
Work Step by Step
Given $tan\theta=\frac{12}{5}\gt0$ and $sin\theta\lt0$, we know $\theta$ is in quadrant III, let $y=-12, x=-5$, we have $r=\sqrt {(-12)^2+(-5)^2}=13$, thus:
1. $sin\theta=\frac{y}{r}=-\frac{12}{13}$,
2. $cos\theta=\frac{x}{r}=-\frac{5}{13}$,
3. $cot\theta=\frac{1}{tan\theta}=\frac{5}{12}$,
4. $sec\theta=\frac{1}{cos\theta}=-\frac{13}{5}$,
5. $csc\theta=\frac{1}{sin\theta}=-\frac{13}{12}$.
