Question

$\alpha =\sin 36^{\circ }$ is a root of which of the following equation

• A. $10x^4-10x^2-5=0$
• B. $16x^4+20x^2-5=0$
• C. $16x^4-20x^2+5=0$
• D. $16x^4-10x^2+5=0$

Answer 1

Answer: (C)

$\cos 72^{\circ }=\frac{\sqrt{5}-1}{4}$
$\Rightarrow 1-2{\sin }^{2}36^{\circ }=\frac{\sqrt{5}-1}{4}$
$\Rightarrow 4-8{\alpha }^2=\sqrt{5}-1$
$\Rightarrow 5-8{\alpha }^2=\sqrt{5}$
$\Rightarrow {\left(5-8{\alpha }^2\right)}^2=5$
$\Rightarrow 25+64{\alpha }^4-80{\alpha }^2=5$
$\Rightarrow 64{\alpha }^4-80{\alpha }^2+20=0$
$\Rightarrow 16{\alpha }^4-20{\alpha }^2+5=0$