# Answer each of the following questions in one word or one sentence or as per exact requirement of the question.

Question:

Answer each of the following questions in one word or one sentence or as per exact requirement of the question.

In a $\triangle \mathrm{ABC}$, if $\sin A$ and $\sin B$ are the roots of the equation $c^{2} x^{2}-c(a+b) x+a b=0$, then find $\angle C .$

Solution:

It is given that $\sin A$ and $\sin B$ are the roots of the equation $c^{2} x^{2}-c(a+b) x+a b=0$.

$\therefore \sin A+\sin B=-\frac{-c(a+b)}{c^{2}} \quad$ (Sum of roots $=-\frac{b}{a}$ )

$\Rightarrow \sin A+\sin B=\frac{a+b}{c}$

$\Rightarrow \sin A+\sin B=\frac{k \sin A+k \sin B}{k \sin C}$   (Sine rule)

$\Rightarrow \sin A+\sin B=\frac{\sin A+\sin B}{\sin C}$

$\Rightarrow \sin C=1=\sin 90^{\circ}$

$\Rightarrow C=90^{\circ}$