Answer
$sin\theta =\frac{5\sqrt {89}}{89}\\ cos\theta =-\frac{8\sqrt {89}}{89}\\ cot\theta =-\frac{8}{5}\\ sec\theta =-\frac{\sqrt {89}}{8}\\ csc\theta =\frac{\sqrt {89}}{5}$
Work Step by Step
1. Given $tan\theta=-\frac{5}{8}$ ($\theta$ in quadrant II), let $x=-8, y=5$, we have $r=\sqrt {64+25}=\sqrt {89}$.
2. Thus
$sin\theta=\frac{y}{r}=\frac{5\sqrt {89}}{89}$,
$cos\theta=\frac{x}{r}=-\frac{8\sqrt {89}}{89}$,
$cot\theta=\frac{x}{y}=-\frac{8}{5}$,
$sec\theta=\frac{r}{x}=-\frac{\sqrt {89}}{8}$,
$csc\theta=\frac{r}{y}=\frac{\sqrt {89}}{5}$
