Answer
$sin ~\theta = -\frac{7}{\sqrt{53}}$
$cos ~\theta =\frac{2}{\sqrt{53}}$
$tan ~\theta =-\frac{7}{2}$
$csc ~\theta = -\frac{\sqrt{53}}{7}$
$sec ~\theta = \frac{\sqrt{53}}{2}$
$cot ~\theta = -\frac{2}{7}$
Work Step by Step
$x = 2,$
$y = -7$
We can form a triangle with the three points $(0,0),(2,0),$ and $(2,-7)$
We can find the length $r$ of the hypotenuse:
$r = \sqrt{(2)^2+(-7)^2} = \sqrt{53}$
Note that the trigonometric values of $\theta$ are equal to the trigonometric values of the angle $\theta-360^{\circ}$
We can find the values of the six trigonometric functions:
$sin ~\theta = \frac{y}{r} = -\frac{7}{\sqrt{53}}$
$cos ~\theta = \frac{x}{r} = \frac{2}{\sqrt{53}}$
$tan ~\theta = \frac{y}{x} = -\frac{7}{2}$
$csc ~\theta = \frac{r}{y} = -\frac{\sqrt{53}}{7}$
$sec ~\theta = \frac{r}{x} = \frac{\sqrt{53}}{2}$
$cot ~\theta = \frac{x}{y} = -\frac{2}{7}$
