Question

In $\Delta ABC$, if $sinA$ and $sinB$ are the roots of the equation $c^2{x}^2-c(a+b)x+ab=0$ then $sinC=??$

• A. $0$
• B. $\frac{1}{2}$
• C. $1\sqrt{2}$
• D. $1$

Answer 1

The correct option is D $1$

We have,

$c^2{x}^2-c(a+b)x+ab=0$

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

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

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

$\Rightarrow cx-a=0,cx-b=0$

$\Rightarrow x=\frac{a}{c}$ or $x=\frac{b}{c}$

As, $\sin A=\frac{a}{c}$ and $\sin B=\frac{b}{c}$

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

$\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}=c$ ........$\left(1\right)$

Using sine law,

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ ..........$\left(2\right)$

So,$\frac{c}{\sin C}=c$ from $\left(1\right)$ and $\left(2\right)$

$\therefore \sin C=1$