Question

Given that cosine of θ equals 4/5 and that θ lies in quadrant IV, what is the exact value of sin 2θ?

a) -24/25
b) -7/25
c) 24/25
d) 7/25

Answer 1

To find the exact value of sin 2θ when θ is in the fourth quadrant and cos θ = 4/5, use the double-angle formula and the Pythagorean identity. The calculation results in sin 2θ being -24/25.

The question asks for the exact value of sin 2θ given that cos θ = 4/5 and θ is in the fourth quadrant. To solve for sin 2θ, we use the double-angle formula for sine, which is sin 2θ = 2sin θ cos θ. Since θ is in the fourth quadrant, the sine value will be negative because sine is negative in the fourth quadrant.

First, we find sin θ using the Pythagorean identity sin² θ + cos² θ = 1. If cos θ = 4/5, then sin θ = -√(1 - (4/5)²) = -√(1 - 16/25) = -√(9/25) = -3/5. Now, applying the double-angle formula, sin 2θ = 2(-3/5)(4/5) = -24/25.