Question

If \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], where A and B both lie in second quadrant, find the value of sin (A + B).

Answer 1

 Given:
\[ \sin A = \frac{3}{5}\text{ and }\cos B = - \frac{12}{13}\]
and that A and B both lie in second qudrant .
We know that in second quadrant sine function is positive and \cosine function is negative .
Therefore,
\[ \cos A = - \sqrt{1 - \sin^2 A}\text{ and }\sin B = \sqrt{1 - \cos^2 B} \]
\[ \Rightarrow \cos A = - \sqrt{1 - \left( \frac{3}{5} \right)^2} \text{ and }\sin B = \sqrt{1 - \left( \frac{- 12}{13} \right)^2} \]
\[ \Rightarrow \cos A = - \sqrt{1 - \frac{9}{25}}\text{ and }\sin B = \sqrt{1 - \frac{144}{169}}\]
\[ \Rightarrow \cos A = - \sqrt{\frac{16}{25}}\text{ and }\sin B = \sqrt{\frac{25}{69}}\]
\[ \Rightarrow \cos A = \frac{- 4}{5}\text{ and }\sin B = \frac{5}{13}\]
Now, 
\[\sin\left( A + B \right) = \sin A \cos B + \cos A \sin B\]
\[ = \frac{3}{5} \times \frac{- 12}{13} + \frac{- 4}{5} \times \frac{5}{13}\]
\[ = \frac{- 36}{65} - \frac{20}{65}\]
\[ = \frac{- 56}{65}\]