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)

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