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 coefficient of $x$ is

• A. $\frac{3}{2}$
• B. $-\frac{3}{2}$
• C. $-\frac{3}{4}$
• D. zero

Answer 1

Answer: (C)

$\cos 36^{\circ} \cdot \cos 84^{\circ}+\cos 84^{\circ} \cos 156^{\circ}+\cos 36^{\circ} \cos 156^{\circ} $
$\theta=24^{\circ}$
$\cos \left(60^{\circ}-\theta\right) \cos \left(60^{\circ}+\theta\right)-\cos \left(60^{\circ}+\theta\right) \cos \theta-\cos (60-\theta) \cos \theta $
$\cos ^2 60^{\circ}-\sin ^2 \theta-\cos \theta\left(\cos \left(60^{\circ}+\theta\right)+\cos \left(60^{\circ}-\theta\right)\right) $
$\cos ^2 60^{\circ}-\sin ^2 \theta-2 \cos \theta \cos 60^{\circ} \cos \theta $
$\cos ^2 60^{\circ}-1=\frac{1}{4}-1=-\frac{3}{4} $