Given sine, theta, equals, minus, start fraction, square root of, 15, end square root, divided by, 4, end fractionsinθ=− 4 15 ​ ​ and angle thetaθ is in Quadrant IV, what is the exact value of cosine, thetacosθ in simplest form? Simplify all radicals if needed.

Answer 1

To find the exact value of cosine θ, we can use the Pythagorean identity:

sin^2θ + cos^2θ = 1

Substituting the given value of sinθ = -4/15:

(-4/15)^2 + cos^2θ = 1

16/225 + cos^2θ = 1

cos^2θ = 1 - 16/225

cos^2θ = (225/225) - (16/225)

cos^2θ = 209/225

cosθ = ±√(209/225)

Since angle θ is in Quadrant IV, cosine is positive in Quadrant IV. Therefore:

cosθ = √(209/225)