Q. If sin $A = \frac{4}{5}\$and $\cos{B = -}\frac{12}{13},\$where $A\$and $B$ lie in first and third quadrant respectively, then cos($A\ + \ B$) is equal to
A) $\frac{56}{65}$
B) $- \frac{56}{65}$
C) $\frac{16}{65}$
D) $- \frac{16}{65}$
Explanation
(d)
We have, $\sin{A =}\frac{4}{5}\ and\cos{B = - \frac{12}{13}}$
Now$,\cos{(A + B) = \cos{A\cos{B - \sin{A\sin B}}}}$
$= \sqrt{1 - \frac{16}{25}}\ \left( - \frac{12}{13} \right) - \frac{4}{5}\ \sqrt{1 - \frac{144}{169}}$
$= - \frac{3}{5} \times \frac{12}{13} - \frac{4}{5}\left( - \frac{5}{13} \right)$
$= - \frac{36}{65} + \frac{20}{65} = - \frac{16}{65}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.