Question

If $\theta$ satisfying
$4{\sin }^2\theta -2(\sqrt{3}+1)\sin \theta +\sqrt{3}=0$, then acute angles A and B are ________________.

• A. $30,45$
• B. $60,30$
• C. $60,90$
• D. $45,60$

Answer 1

The correct option is D $45,60$

Let $\sin A,\sin B$ be the roots of the equation.

Sum of the roots$=\sin A+\sin B=\frac{2(\sqrt{3}+1)}{4}=\frac{\sqrt{3}}{2}+\frac{1}{2}$

$\Rightarrow \sin A+\sin B=\frac{\sqrt{3}}{2}+\frac{1}{2}=\sin {60}^{\circ }+\sin {30}^{\circ }$

Product of the roots$=\sin A\sin B=\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\times \frac{1}{2}=\sin {60}^{\circ }\sin {30}^{\circ }$

$\therefore \sin A=\sin {60}^{\circ },\sin B=\sin {30}^{\circ }$

Hence $A={60}^{\circ }$ and $B={30}^{\circ }$