Finding Csc from point?

Question

How do I find csc theta given that (9,12) is a point on the terminal side of the angle theta?

How do I do this? Thanks.

Answer 1

The length of the line segment connecting (0,0) to (9,12) is sqrt(9² + 12²) = sqrt(225) = 15. So the intersection of the terminal side of the angle with the unit circle is (9/15, 12/15) or after reducing the fractions, (3/5, 4/5). From these we know that cos(theta) = 3/5 and sin(theta) = 4/5. Since csc(theta) = 1/sin(theta), we have csc(theta) = 5/4.