Question

Consider the polynomial $P ( x )=\left( x -\cos 36^{\circ}\right)\left( x -\cos 84^{\circ}\right)\left( x -\cos 156^{\circ}\right)$
The absolute term in $P ( x )$ has the value equal to

• A. $\frac{\sqrt{5}-1}{4}$
• B. $\frac{\sqrt{5}-1}{16}$
• C. $\frac{\sqrt{5}+1}{16}$
• D. $\frac{1}{16}$

Answer 1

Answer: (B)

$\text { Using }-\cos 36^{\circ} \cos 84^{\circ} \cos 156^{\circ} $
$\cos 24^{\circ} \cos 36^{\circ} \cos 84^{\circ} $
$\cos \theta \cos \left(60^{\circ}-\theta\right) \cos \left(60^{\circ}-\theta\right)=\frac{\cos 3 \theta}{4}$
$=\frac{\cos (3 \cdot 24)}{4}=\frac{\cos 72^{\circ}}{4}=\frac{\sqrt{5}-1}{4 \cdot 4}=\frac{\sqrt{5}-1}{16}$