Find the exact value of cos(alpha + beta) given that sin alpha = 2/3 and sin beta = -1/2 with alpha in quadrant 1 and beta in quadrant 3
Since sin(alpha) = 2/3 and alpha is in quadrant 1, we can use the Pythagorean identity to find cos(alpha):
cos(alpha) = √(1 - sin^2(alpha))
cos(alpha) = √(1 - (2/3)^2)
cos(alpha) = √(1 - 4/9)
cos(alpha) = √(5/9)
cos(alpha) = √5 / 3
Similarly, since sin(beta) = -1/2 and beta is in quadrant 3, we can also find cos(beta) using the Pythagorean identity:
cos(beta) = -√(1 - sin^2(beta))
cos(beta) = -√(1 - (-1/2)^2)
cos(beta) = -√(1 - 1/4)
cos(beta) = -√3/2
Now, to find cos(alpha + beta), we can use the sum formula for cosine:
cos(alpha + beta) = cos(alpha)cos(beta) - sin(alpha)sin(beta)
cos(alpha + beta) = (√5 / 3)(-√3 / 2) - (2/3)(-1/2)
cos(alpha + beta) = -√15 / 6 + 1/3
cos(alpha + beta) = (1 - √15) / 3
Therefore, the exact value of cos(alpha + beta) is (1 - √15) / 3.