Q. If $\sin{\theta = \frac{12}{13},\ }\ \left( 0 < \theta < \frac{\pi}{2} \right)$ and $\cos{\phi = - \frac{3}{5}}\left( \pi < \phi < \frac{3\ \pi}{2} \right)$, then $\sin{(\theta + \phi)}$ will be
A) $- 56/61$
B) $- 56/65$
C) $1/65$
D) $- 56$
Explanation
(b)
Given, $\sin{\theta = \frac{12}{13}\ }$ and $\cos{\phi = - \frac{3}{5}}$
$\therefore\ \cos{\theta = \frac{5}{13}}$ and $\sin{\phi = - \frac{4}{5}}$
$\therefore\ \sin{(\theta + \phi) = \sin{\theta\cos{\phi + \cos{\theta\sin\phi}}}}$
$= \frac{12}{13} \times \left( - \frac{3}{5} \right) + \frac{5}{13} \times \left( - \frac{4}{5} \right)$
$= \frac{- 36}{65} + \frac{( - 20)}{65} = - \frac{56}{65}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.