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, $\cos 27^{\circ}-\sin 27^{\circ}=$
$\sqrt{1-\cos 36^{\circ}}\left(\right.$ since, $\left.\cos 27^{\circ}>\sin 27^{\circ}\right)$
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 $\beta=3-\sqrt{5}$
$\Rightarrow \alpha+\beta=8$