Question

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.

Answer 1

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.