Given sine, theta, equals, minus, one halfsinθ=−
2
1
and angle thetaθ is in Quadrant IV, 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 value of cosine(theta), we can use the Pythagorean Identity, which states that sin^2(theta) + cos^2(theta) = 1.
Since sin(theta) = -1/2 in Quadrant IV, we can determine cos(theta) as follows:
sin^2(theta) + cos^2(theta) = 1
(-1/2)^2 + cos^2(theta) = 1
1/4 + cos^2(theta) = 1
cos^2(theta) = 1 - 1/4
cos^2(theta) = 3/4
Taking the square root of both sides gives us:
cos(theta) = sqrt(3/4)
Since theta is in Quadrant IV, the value of cosine(theta) is positive. Simplifying the radical gives us:
cos(theta) = sqrt(3)/sqrt(4) = sqrt(3)/2
So the exact value of cosine(theta) is sqrt(3)/2.