If sin A = 3over5 and cos B = 12over13 , and A and B ∈ (0, (π)/2) , find sin(A + B)

If sin A = 3over5 and cos B = 12over13 , and A and B ∈ (0…

If $\sin A = {3\over5}$ and $\cos B = {12\over13}$ , and A and $B \in \left(0, \frac{\pi}{2}\right)$ , find sin(A + B)

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Correct answer: A

Explanation

The correct answer is Option 1 i.e. $56\over 65$ .

$\cos^2 A = 1 - \sin^2 A = 1 - \frac{9}{25} = \frac{16}{25}$

⇒ cos A = 4/5, Positive since $A \in (0, \pi/2)$

$\sin^2 B = 1 - \cos^2 B = 1 - \frac{144}{169} = \frac{25}{169}$

⇒ sin B = 5/13, Positive since $B \in (0, \pi/2)$

sin(A + B) = sin A cos B + cos A sin B

⇒ $\sin(A + B) = \frac{3}{5} \times \frac{12}{13} + \frac{4}{5} \times \frac{5}{13}= \frac{36}{65} + \frac{20}{65} = \frac{56}{65}$