Question

In $\Delta ABC$ If $\sin A$ and $\sin B$ are the roots of the equation $c^2{x}^2-c\left(a+b\right)x+ab=0$, then

Answer 1

Consider the given equation.

$c^2{x}^2-(a+b)x+ab=0$ ……. (1)

$c^2{x}^2-cax-cbx+ab=0$

$cx(cx-a)-b(cx-a)=0$

$(cx-b)(cx-a)=0$

$x=\frac{b}{c},\frac{a}{c}$

Since, $\sin A,sinB$ are the roots of above equation

Let $sinA=\frac{b}{c},sinB=\frac{a}{c}$

$\Rightarrow c=\frac{a}{\sin A}=\frac{b}{\sin B}$ …….. (2)

We know that

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

So,

$\frac{\mathrm{sinC}}{C}=\frac{1}{C}$

$\sin C=1$

$\sin C=\sin \frac{\pi }{2}$

$C=\frac{\pi }{2}$

Hence, this is the answer.