Answer
$sin ~\theta = \frac{\sqrt{2}}{2}$
$cos ~\theta = \frac{-\sqrt{2}}{2}$
$tan ~\theta = -1$
$csc ~\theta = \sqrt{2}$
$sec ~\theta = -\sqrt{2}$
$cot ~\theta = -1$
Work Step by Step
$135^{\circ}= 180^{\circ}-45^{\circ}$
The angle $135^{\circ}$ makes an angle of $45^{\circ}$ above the negative x-axis. We can let $x=-1$ and $y=1$. Then $r = \sqrt{2}$.
We can find the values of the six trigonometric functions:
$sin ~\theta = \frac{y}{r} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
$cos ~\theta = \frac{x}{r} = \frac{-1}{\sqrt{2}} = \frac{-\sqrt{2}}{2}$
$tan ~\theta = \frac{y}{x} = \frac{1}{-1} = -1$
$csc ~\theta = \frac{r}{y} = \frac{\sqrt{2}}{1} = \sqrt{2}$
$sec ~\theta = \frac{r}{x} = \frac{\sqrt{2}}{-1} = -\sqrt{2}$
$cot ~\theta = \frac{x}{y} = \frac{-1}{1} = -1$
