Answer
$\sin \theta=\frac{4}{5}$
$\cos \theta=\frac{-3}{5}$
$\tan \theta=\frac{4}{-3}$
$\csc \theta=\frac{5}{4}$
$\sec \theta=\frac{5}{-3}$
$\cot \theta=\frac{-3}{4}$
Work Step by Step
We are given $x=-3, y=4$
x,y,z form a right-angled triangle so to find z we can apply:
$x^2+y^2=z^2$
$\rightarrow z=\sqrt (-3)^2+4^2=5$
With $x=-3, y=4, z=5$, the trigonometric functions are
$\sin \theta=\frac{y}{z}=\frac{4}{5}$
$\cos \theta=\frac{x}{z}=\frac{-3}{5}$
$\tan \theta=\frac{y}{x}=\frac{4}{-3}$
$\csc \theta=\frac{1}{\sin \theta}=\frac{1}{\frac{4}{5}}=\frac{5}{4}$
$\sec \theta=\frac{1}{\cos \theta}=\frac{1}{\frac{-3}{5}}=\frac{5}{-3}$
$\cot \theta=\frac{x}{y}=\frac{-3}{4}$
