Garrett C. answered • 08/31/21
Trigonometry Teacher of a Self-Designed Elective Course
An angle terminates in the 4th Quadrant and has cosecant equal to -13/5. What is it’s cotangent?
We must begin this by first creating the triangle in the 4th quadrant and labeling all necessary information. Since we are given that the csc(x) = 13/-5, this means its reciprocal is sine, so sin(x) = -5/13.
This lets us know that the vertical side of this 4th quadrant triangle is -5, and the hypotenuse length is 13. We must find the horizontal length of the triangle. Since we are in the 4th quadrant, this length is positive as this side is on the positive x-axis. Either by using the Pythagorean Theorem, OR by knowing about Pythagorean Triplets, we can find this horizontal length to be 12.
Cotangent is the reciprocal of tangent, and the tangent ratio for this triangle is opposite/adjacent, or -5/12.
So tan(x) = -5/12, and therefore cot(x) = 12/-5.
