Question

Find $a$ if, $a\sin {27}^0={(5+\sqrt{5})}^{1/2}-{(3-\sqrt{5})}^{1/2}$

Answer 1

Consier, $16{\sin }^2{27}^0$

$=8(1-\cos {54}^0)$

$=8(1-\frac{\sqrt{10}-2\sqrt{5}}{4})$

$=2(4-\sqrt{10}-2\sqrt{5})$

$=8-2\sqrt{10}-2\sqrt{5}$

$=(5+\sqrt{5})+(3-\sqrt{5})-2\sqrt{(5+\sqrt{5})(3-\sqrt{5})}$

$={\{\sqrt{5+\sqrt{5}}-\sqrt{3-\sqrt{5}}\}}^2$

$\Rightarrow$ $4\sin {27}^0=\sqrt{5+\sqrt{5}}-\sqrt{3-\sqrt{5}}$
Ans: 4