Question

If $\alpha$ and $\beta$ are the two different roots of the equation $a\cos \theta +b\sin \theta =c$, then show that $\sin \alpha +\sin \beta =\frac{2bc}{a^2+b^2}$ and $\sin \alpha \sin \beta =\frac{c^2-a^2}{a^2+b^2}$.

Answer 1

$a\cos \theta =c-b\sin \theta \Rightarrow a^2(1-{\sin }^2\theta )=c^2-2bc\sin \theta +b^2{\sin }^2\theta \Rightarrow (a+b^2){\sin }^2\theta -2bc\sin \theta +(c^2+a^2)=0$

Therefore roots of the above equation are $\sin \alpha$ and $\sin \beta$

$\therefore \sin \alpha +\sin \beta =\frac{2bc}{a^2+b^2}$ and $\sin \alpha \sin \beta =\frac{c^2-a^2}{a^2+b^2}$