Find the exact value of sin(alpha + beta) given that sin alpha = 3/5 and sin beta = 5/13 with alpha in quadrant 2 and beta in quadrant 1
To find the exact value of sin(alpha + beta), we can use the sum formula for sine:
sin(alpha + beta) = sin(alpha) * cos(beta) + cos(alpha) * sin(beta)
First, let's find the values of cos(alpha) and cos(beta) using the Pythagorean identity:
cos^2(alpha) = 1 - sin^2(alpha)
cos^2(alpha) = 1 - (3/5)^2
cos^2(alpha) = 1 - 9/25
cos^2(alpha) = 16/25
cos(alpha) = ±√(16/25) = ±4/5
Since alpha is in quadrant 2, cos(alpha) is negative, so cos(alpha) = -4/5.
Similarly, let's find cos(beta):
cos^2(beta) = 1 - sin^2(beta)
cos^2(beta) = 1 - (5/13)^2
cos^2(beta) = 1 - 25/169
cos^2(beta) = 144/169
cos(beta) = ±√(144/169) = ±12/13
Since beta is in quadrant 1, cos(beta) is positive, so cos(beta) = 12/13.
Now, substitute these values into the sum formula for sine:
sin(alpha + beta) = sin(alpha) * cos(beta) + cos(alpha) * sin(beta)
sin(alpha + beta) = (3/5) * (12/13) + (-4/5) * (5/13)
sin(alpha + beta) = 36/65 - 20/65
sin(alpha + beta) = 16/65
Therefore, the exact value of sin(alpha + beta) is 16/65.