Given sine, theta, equals, start fraction, square root of, 21, end square root, divided by, 5, end fractionsinθ=
5
21
and angle thetaθ is in Quadrant II, what is the exact value of cosine, thetacosθ in simplest form? Simplify all radicals if needed.
In Quadrant II, the cosine of an angle is negative.
Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can solve for cosθ:
sin^2θ + cos^2θ = 1
cos^2θ = 1 - sin^2θ
cos^2θ = 1 - (21/25)
cos^2θ = 4/25
cosθ = ± sqrt(4/25)
cosθ = ± (2/5)
Since angle thetaθ is in Quadrant II, cosine is negative. Therefore,
cosθ = -2/5.
So, the exact value of cosine of theta in simplest form is -2/5.