Question

In a $\Delta leABC$, $A$ is an obtuse angle.If $SinA=3/5,SinB=5/13$ then $SinC=$

• A. $\frac{56}{65}$
• B. $\frac{10}{65}$
• C. $\frac{16}{65}$
• D. $\frac{17}{65}$

Answer 1

The correct option is A $\frac{56}{65}$

Given:$\sin A=\frac{oppositeside}{hypotenuse}=\frac{3}{5}$

$\Rightarrow adjacentside=\sqrt{{\left(hypotenuse\right)}^2-{\left(oppositeside\right)}^2}$

$=\sqrt{5^2-3^2}=\sqrt{25-9}=\sqrt{16}=4$

$\Rightarrow \cos A=\frac{adjacentside}{hypotenuse}=\frac{4}{5}$

$\therefore \sin A=\frac{3}{5},\cos A=\frac{4}{5}$

Given:$\sin B=\frac{oppositeside}{hypotenuse}=\frac{5}{13}$

$\Rightarrow adjacentside=\sqrt{{\left(hypotenuse\right)}^2-{\left(oppositeside\right)}^2}$

$=\sqrt{{13}^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$

$\Rightarrow \cos B=\frac{adjacentside}{hypotenuse}=\frac{12}{13}$

$\therefore \sin B=\frac{5}{13},\cos B=\frac{12}{13}$

We know that $A+B+C=\pi$

$\Rightarrow A+B=\pi -C$

$\Rightarrow \sin (A+B)=\sin (\pi -C)$

$\Rightarrow \sin A\cos B+\cos A\sin B=\sin C$

$\Rightarrow \frac{3}{5}\times \frac{12}{13}+\frac{4}{5}\times \frac{5}{13}=\sin C$

$\Rightarrow \sin C=\frac{36+20}{65}=\frac{56}{65}$