If \(\sin A = \dfrac{3}{5}\) and \(\cos B = \dfrac{12}{13}\), A and B ∈ \(\left(0, \dfrac{\pi}{2}\right)\), find \(\sin(A+B)\).
\(\dfrac{56}{65}\)
From \(\sin A = \frac{3}{5}\) and first quadrant: \(\cos A = \frac{4}{5}\)
From \(\cos B = \frac{12}{13}\) and first quadrant: \(\sin B = \frac{5}{13}\)
\(\sin(A+B) = \sin A\cos B + \cos A\sin B\)
\(= \dfrac{3}{5} \cdot \dfrac{12}{13} + \dfrac{4}{5} \cdot \dfrac{5}{13} = \dfrac{36}{65} + \dfrac{20}{65} = \mathbf{\dfrac{56}{65}}\)
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