Find csc θ and cot θ, given that sec θ = 5/4 and csc θ < 0.

Question

csc θ = ?

cot θ = ?

Answer 1

If sec θ > 0 (or x > 0) and csc θ < 0 (or y <0) then the angle is found in the 4th quadrant.

If sec θ = adjacent/ opposite = r/x = 5/4.

To find y, you can use the Pythagorean Theorem.

r² = x² + y²

5² = 4² + y²

5² - 4² = y²

y² = 25 - 16 = 9

y = ±3

Remember, the angle θ is located in the 4th quadrant

∴ y = -3

csc θ = r / y = -5/3

cot θ = x / y = -4/3