Question

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

In a ∆ABC, if sinA and sinB are the roots of the equation $c^2{x}^2-c\left(a+b\right)x+ab=0$, then find $\angle C$.

Answer 1

It is given that sinA and sinB are the roots of the equation $c^2{x}^2-c\left(a+b\right)x+ab=0$.

$$\begin{gathered} \therefore \sin A+\sin B=-\frac{-c\left(a+b\right)}{c^2}\left(\mathrm{Sum}\mathrm{of}\mathrm{roots}=-\frac{b}{a}\right) \\ \Rightarrow \sin A+\sin B=\frac{a+b}{c} \\ \Rightarrow \sin A+\sin B=\frac{k\sin A+k\sin B}{k\sin C}\left(\mathrm{Sine}\mathrm{rule}\right) \end{gathered}$$
$$\begin{gathered} \Rightarrow \sin A+\sin B=\frac{\sin A+\sin B}{\sin C} \\ \Rightarrow \sin C=1=\sin 90° \\ \Rightarrow C=90° \end{gathered}$$