Let θ be an angle in quadrant IV such that sinθ=−3/5 . Find the exact values of secθ and cotθ .

Question

Let θ be an angle in quadrant IV such that sinθ=−3/5 .

Find the exact values of secθ and cotθ .

Answer 1

I'm going to use x vice theta because it's easier to type.  You will need to use the following identities"

 

cos²(x) + sin²(x) = 1

sec(x) = 1/cos(x)

cot(x) = cos(x)/sin(x)

 

sin(x) = -3/5

 

cos²(x) + sin²(x) = 1

cos²(x) = 1 - sin²(x)

cos²(x) = 1 - 9/25 = 16/25

cos(x) = ±√(16/25) = 4/5  (cosine is positive in Q IV)   (3-4-5 right triangle!)

 

sec(x) = 1/cos(x) = 5/4

 

cot(x) = cos(x)/sin(x) = (5/4)/(-3/5) = -4/3

Answer 2

The given data tells us that the cosine must be 4/5 because sin²θ + cos²θ = 1.

From this information, you can easily find all other trig function values of this angle.