Question

Find csc θ if cot θ = −21/20 and sin θ > 0.

Answer 1

Note the answer must be positive because if sin>0, then csc > 0.
There are two ways to do this problem.
1) Use a trig identity:
$csc^2 \theta = cot^2 \theta + 1$
Sub in value for cot
$csc^2 \theta = (-\frac{21}{20})^2 + 1 \\ \\ csc^2 \theta = \frac{841}{400} \\ \\ csc \theta = \frac{\sqrt{841}}{\sqrt{400}} = \frac{29}{20}$

2) Use a right triangle and SOH CAH TOA
cot = 1/tan = adjacent/opposite = -21/20
Therefore adjacent side = 21, opposite side = 20
Use pythagorean thm to find hypotenuse:
$\sqrt{21^2 + 20^2} = \sqrt{841} = 29$

csc = 1/sin = hypotenuse/opposite = 29/20

Hope this helps.