Given cosine, theta, equals, start fraction, 2, divided by, 3, end fractioncosθ=
3
2
and angle thetaθ is in Quadrant I, 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=
To find the value of sine(theta), we can use the Pythagorean identity:
cos^2(theta) + sin^2(theta) = 1
Since we are given that cos(theta) = 2/3, we can square this value and substitute it into the equation above:
(2/3)^2 + sin^2(theta) = 1
4/9 + sin^2(theta) = 1
sin^2(theta) = 1 - 4/9
sin^2(theta) = 5/9
Taking the square root of both sides, we get:
sin(theta) = ±sqrt(5)/3
Since theta is in Quadrant I and sin(theta) is positive in Quadrant I, the exact value of sin(theta) is sqrt(5)/3.