Solveeit Logo
Login

Question

Question: If \(\sin A = \frac{4}{5}\)and \(\cos B = - \frac{12}{13},\) where A and B lie in first and third qu...

If sin⁡A=45\sin A = \frac{4}{5}and cos⁡B=−1213,\cos B = - \frac{12}{13}, where A and B lie in first and third quadrant respectively, then cos⁡(A+B)=\cos(A + B) =

A

5665\frac{56}{65}

B

−5665- \frac{56}{65}

C

1665\frac{16}{65}

D

– 1665\frac{16}{65}

Answer

– 1665\frac{16}{65}

Explanation

Solution

We have sin⁡A=45\sin A = \frac{4}{5} and cos⁡B=−1213\cos B = - \frac{12}{13}

Now, cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A\cos B - \sin A\sin B

=1−1625(−1213)−451−144169= \sqrt{1 - \frac{16}{25}}\left( - \frac{12}{13} \right) - \frac{4}{5}\sqrt{1 - \frac{144}{169}}

=−35×1213−45(−513)=−1665= - \frac{3}{5} \times \frac{12}{13} - \frac{4}{5}\left( - \frac{5}{13} \right) = - \frac{16}{65}

(Since A lies in first quadrant and B lies in third quadrant).