If \( \sin A = \frac{3}{5} \) and \( \cos B = \frac{15}{17} \), where A and B are acute angles, find the exact value of \( \sin(A+B) \).

Steps to Solve

1. Find $\cos A$ using the Pythagorean identity

Since $\sin A = \frac{3}{5}$ and $A$ is an acute angle, we can use the Pythagorean identity $\sin^2 A + \cos^2 A = 1$ to find $\cos A$: $$ \cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} $$ Since $A$ is an acute angle, $\cos A$ is positive. Thus, $$ \cos A = \sqrt{\frac{16}{25}} = \frac{4}{5} $$

2. Find $\sin B$ using the Pythagorean identity

Since $\cos B = \frac{15}{17}$ and $B$ is an acute angle, we can use the Pythagorean identity $\sin^2 B + \cos^2 B = 1$ to find $\sin B$: $$ \sin^2 B = 1 - \cos^2 B = 1 - \left(\frac{15}{17}\right)^2 = 1 - \frac{225}{289} = \frac{64}{289} $$ Since $B$ is an acute angle, $\sin B$ is positive. Thus, $$ \sin B = \sqrt{\frac{64}{289}} = \frac{8}{17} $$

3. Apply the sine addition formula

Now, we can use the formula for $\sin(A+B)$: $$ \sin(A+B) = \sin A \cos B + \cos A \sin B $$ Substitute the given values: $$ \sin(A+B) = \left(\frac{3}{5}\right)\left(\frac{15}{17}\right) + \left(\frac{4}{5}\right)\left(\frac{8}{17}\right) = \frac{45}{85} + \frac{32}{85} = \frac{77}{85} $$