Given sine, theta, equals, one quartersinθ=
4
1
and angle thetaθ is in Quadrant I, what is the exact value of cosine, thetacosθ in simplest form? Simplify all radicals if needed.
Answer
Attempt 1 out of 2
cosine, theta, equalscosθ=
To find the exact value of cosine theta (cosθ), we can use the Pythagorean identity:
sin^2θ + cos^2θ = 1
Since we are given sinθ = 1/4, we can square it to find sin^2θ:
(1/4)^2 = 1/16
Now we can substitute the value of sin^2θ into the Pythagorean identity:
1/16 + cos^2θ = 1
To isolate cos^2θ, we can subtract 1/16 from both sides:
cos^2θ = 1 - 1/16
cos^2θ = 16/16 - 1/16
cos^2θ = 15/16
Now we can take the square root of both sides to find the value of cosθ:
cosθ = ±√(15/16)
Since the angle theta is in Quadrant I, cosine theta will be positive:
cosθ = √(15/16)
Simplifying the radical by canceling out the perfect square:
cosθ = √15/√16
Finally, we can simplify the radical further:
cosθ = (√15)/(√16)
cosθ = (√15)/4
Therefore, the exact value of cosine theta (cosθ) is (√15)/4.