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)
$\Rightarrow \sin A+\sin B=\frac{\sin A+\sin B}{\sin C}$
$\Rightarrow \sin C=1=\sin 90^{\circ}$
$\Rightarrow C=90^{\circ}$