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.

Question

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)

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.

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

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.