TITLE: Finding $ \csc \theta $ given $ \cot \theta $
QUESTION [7 upvotes]: I have the following problem:
If $ \cot{C} = \frac{\sqrt{3}}{7} $, find $ \csc{C} $
From my trig identities, I know that $ \cot{\theta} = \frac{1}{\tan{\theta}} $, and $ \csc{\theta} = \frac{1}{\sin{\theta}} $, and also $ \cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} $
However, I can't seem to see how to connect the dots to get from cotangent to cosecant. I figure I might be able to use the last identity if I can somehow make $ \cos{C} = 1 $, but I don't really see how to do that, either.
This is homework, so please provide me with some pointers rather than complete solutions.
Thanks.

REPLY [3 votes]: Here's a simple way I usually think about it. Suppose you have a right triangle in the first quadrant. Since $\cot C=\frac{\sqrt{3}}{7}$, you know the ratio of the leg adjacent to $C$ to the leg opposite of $C$ is $\frac{\sqrt{3}}{7}$. So let's just say the opposite leg has length $7$ and the adjacent leg has length $\sqrt{3}$. Then by the Pythagorean theorem, the hypotenuse has length $\sqrt{52}=2\sqrt{13}$. 
Now $\csc C$ is just the ratio of the hypotenuse to the opposite leg, essentially the inverse of $\sin$, which is the ratio of the opposite leg to the hypotenuse.