Question

If $\sin \alpha =12/13(0<\alpha <\pi /2)$ and
$\cos \beta =-\frac{3}{5}(\pi <\beta <\frac{3}{2}\pi )$, the value of $\sin (\alpha +\beta )$ is

• A. $\frac{-56}{65}$
• B. $\frac{16}{65}$
• C. $\frac{56}{65}$
• D. $\frac{-16}{65}$

Answer 1

Answer: (A)

$\frac{-56}{65}$

Ans. $\left(a\right)$
Since $0<\alpha <\pi$ we have $\sin \alpha$ =$\frac{12}{13}$
$\Rightarrow \cos \alpha =\surd [1-{\left(12/13\right)}^2]$=$\frac{5}{13}$
And since $\pi <\beta <3\pi /2$, we have $\cos \beta =-3/5\Rightarrow \sin \beta =-4/5$
Hence $\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$
$=\frac{12}{13}(-\frac{3}{5})+\left(\frac{5}{13}\right)(-\frac{4}{5})=-\frac{56}{65}$