Given cosine, theta, equals, start fraction, 2, divided by, 5, end fractioncosθ=
5
2
and angle thetaθ is in Quadrant IV, what is the exact value of sine, thetasinθ in simplest form? Simplify all radicals if needed.
Answer
Attempt 1 out of 2
sine, theta, equalssinθ=
We are given that cosθ = 2/5 and angle θ is in Quadrant IV. In Quadrant IV, both the sine and cosine are positive.
Since cosθ = 2/5, we can use the equation sin^2θ + cos^2θ = 1 to find sinθ.
Plugging in the given value of cosθ, we have:
sin^2θ + (2/5)^2 = 1
sin^2θ + 4/25 = 1
sin^2θ = 1 - 4/25
sin^2θ = 25/25 - 4/25
sin^2θ = 21/25
Taking the square root of both sides, we get:
sinθ = ± √(21/25)
Since θ is in Quadrant IV and both the sine and cosine are positive in Quadrant IV, we can conclude that the exact value of sinθ is √(21/25).
Thus, the exact value of sinθ in simplest form is √21/5.