Question

If $4\sin 27^{\circ }=\sqrt{\alpha }-\sqrt{\beta }$ , then the value of $\alpha +\beta$ is

• A. $5$
• B. $3$
• C. $8$
• D. $2$

Answer 1

Answer: (C)

$\cos 27^{\circ }+\sin 27^{\circ 2}=1+2\sin 27^{\circ }$
$\cos 27^{\circ }=1+\sin 54^{\circ }=1+\cos 36^{\circ }$
$\Rightarrow \cos 27^{\circ }+\sin 27^{\circ }=\sqrt{1+\cos 36^{\circ }}$
Similarly, $\cos 27^{\circ }-\sin 27^{\circ }=$
$\sqrt{1-\cos 36^{\circ }}($ since, $\cos 27^{\circ }>\sin 27^{\circ })$
On subtracting, we get,
$2\sin 27^{\circ }=\sqrt{1+\cos 36^{\circ }}-\sqrt{1-\cos 36^{\circ }}$
$=\sqrt{1+\frac{\sqrt{5}+1}{4}}-\sqrt{1-\frac{\sqrt{5}+1}{4}}$
$=\frac{\sqrt{5+\sqrt{5}}}{2}-\frac{\sqrt{3-\sqrt{5}}}{2}$
$\Rightarrow 4\sin 27^{\circ }=\sqrt{5+\sqrt{5}}-\sqrt{3-\sqrt{5}}$
$\Rightarrow \alpha =5+\sqrt{5}$ and $\beta =3-\sqrt{5}$
$\Rightarrow \alpha +\beta =8$