Find the exact value of s as in(alpha + beta) given that sin alpha = 3/5 and sin b= 5/13 with alpha in quadrant 2 and beta in quadrant 1
To find the exact value of s in (alpha + beta), we can use the sum-to-product identities for sine.
Let's start with finding the cosine values of alpha and beta using the given information.
Given that sin alpha = 3/5 and alpha is in quadrant 2, we can use the Pythagorean identity sin^2(alpha) + cos^2(alpha) = 1 to find cos alpha.
sin^2(alpha) = (3/5)^2 = 9/25
cos^2(alpha) = 1 - sin^2(alpha) = 1 - 9/25 = 16/25
cos(alpha) = ± √(16/25) = ± 4/5
Since alpha is in quadrant 2, where cosine is negative, we have cos(alpha) = -4/5.
Similarly, given that sin beta = 5/13 and beta is in quadrant 1, we use the Pythagorean identity to find cos beta.
sin^2(beta) = (5/13)^2 = 25/169
cos^2(beta) = 1 - sin^2(beta) = 1 - 25/169 = 144/169
cos(beta) = ± √(144/169) = ± 12/13
Since beta is in quadrant 1, where cosine is positive, we have cos(beta) = 12/13.
Now, let's consider the sum-to-product identity for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
In this case, A = alpha and B = beta.
sin(alpha + beta) = sin(alpha)cos(beta) + cos(alpha)sin(beta)
= (3/5)(12/13) + (-4/5)(5/13)
= 36/65 - 20/65
= 16/65
Therefore, the exact value of s in (alpha + beta) is 16/65.