Given sine, theta, equals, minus, start fraction, square root of, 13, end square root, divided by, 7, end fractionsinθ=−
7
13
and angle thetaθ is in Quadrant III, 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 (cosθ), we can use the identity cosθ = ±√(1 - sin^2θ).
Since sine, theta (sinθ) is given as -7/13, we can substitute this value into the identity:
cosθ = ±√(1 - (-7/13)^2)
Simplifying the expression inside the square root:
cosθ = ±√(1 - 49/169)
cosθ = ±√(169/169 - 49/169)
cosθ = ±√(120/169)
Taking the square root:
cosθ = ±√(120)/√(169)
cosθ = ±√(120)/13
Therefore, the exact value of cosine, theta (cosθ), in simplest form is ±√(120)/13.