Let θ be an angle such that cosθ= 5/13 and tanθ < 0.

Question

Find the exact values of cotθ and cscθ

cotθ =

cscθ =

I know the quadrant it’s supposed to be in but I don’t know how to solve it

Answer 1

Hi Babe R.,

When the question says "Exact values" that should be a tip-off that you have a "special" triangle involved, which you do. The angle is in the 4th quadrant, as you indicate you know, and, together with the x-axis, it makes a 5-12-13 side length right triangle. You should recognize the parts of a 3-4-5, a 5-12-13, and perhaps a 7-24-25 Pythagorean triples integer side length triangle. There are infinitely many more; look up "Pythagorean triples" if you are interested.

you can handle the rest of the ratios, I think?

--Cheers, - Mr. d.

Answer 2

cos(θ) = 5/13

hypotenuse = 13

adjacent leg = 5

Calculating the length of the opposite leg:

opposite leg = √(13² - 5²) = 12

sin(θ) = - 12/13 ← the negative sign is because tan(θ) < 0)

csc(θ) = 1/sin(θ) = -13/12

cot(θ) = -5/12