Given cosine, theta, equals, minus, start fraction, square root of, 21, end square root, divided by, 5, end fractioncosθ=−
5
21
and angle thetaθ is in Quadrant III, what is the exact value of sine, thetasinθ in simplest form? Simplify all radicals if needed.
In Quadrant III, the sine function is negative. We can use the Pythagorean Identity to find the value of sine theta.
Since cosine theta is equal to -(√21)/5, we can let the adjacent side be -(√21) and the hypotenuse be 5. By the Pythagorean Theorem, we can find the opposite side.
Let the opposite side be x.
(√21)^2 + x^2 = 5^2
21 + x^2 = 25
x^2 = 4
x = 2
So, the exact value of sine theta, sin(theta), is -2/5.