Question

If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:

cos (A + B)

Answer 1

Given: \[\sin A = \frac{4}{5}\text{ and }\cos B = \frac{5}{13}\]
We know that

\[\cos A = \sqrt{1 - \sin^2 A}\text{and}\sin B = \sqrt{1 - \cos^2 B}, \text{where}0 < A, B < \frac{\pi}{2}\]

\[\Rightarrow \cos A = \sqrt{1 - \left( \frac{4}{5} \right)^2}\text{ and }\sin B = \sqrt{1 - \left(\frac{5}{13} \right)^2}\]

\[\Rightarrow \cos A = \sqrt{1 - \frac{16}{25}}\text{ and }\sin B = \sqrt{1 - \frac{25}{169}}\]

\[\Rightarrow \cos A = \sqrt{\frac{9}{25}}\text{ and }\sin B = \sqrt{\frac{144}{169}}\]

\[\Rightarrow \cos A = \frac{3}{5}\text{and}\sin B = \frac{12}{13}\]
Now,

\[\cos\left( A + B \right) = \cos A \cos B - \sin A \sin B\]

\[ = \frac{3}{5} \times \frac{5}{13} - \frac{4}{5} \times \frac{12}{13}\]

\[ = \frac{15}{65} - \frac{48}{55}\]

\[ = \frac{- 33}{65}\]