Find the exact value of the expression

Question

sin[sin^(-1)(3)/(5)+cos^(-1)(-(5)/(13))]

Answer 1

sin(a+b) = sin a cos b + cos a sin b

As a = sin^(-1)(3)/(5), sin(a) = 3/5 and cos(a) = sqrt(1 - sin^2(a)) = sqrt(16/25) = 4/5 and is positive, as a is in Quadrant 1.

As b = arccos(-5/13), we are in quadrant 2, where sin is positive, so sin(b) = sqrt(1 - (-5/13)^2) = sqrt(1 - 25/169) = sqrt(144/169) = 12/13 (alternatively, use the fact that 5-12-13 is a Pythagorean triple).

Then, sin a cos b + cos a sin b = 3/5 * (-5/13) + 4/5 * 12/13 = -3/13 + 48/65 = 33/65