Question

Find all values of $\theta$ , between 0 & $\pi$, which satisfy the equation $\cos \theta \cos 2\theta \cos 3\theta =\frac{1}{4}$.

• A. $0.\frac{\pi }{8},\frac{\pi }{3},\frac{3\pi }{8},\frac{5\pi }{8},\frac{2\pi }{3},\frac{7\pi }{8},\pi$
• B. $\frac{\pi }{8},\frac{\pi }{3},\frac{3\pi }{8},\frac{5\pi }{8},\frac{2\pi }{3},\frac{7\pi }{8}$
• C. $\frac{\pi }{8},\frac{\pi }{3},\frac{3\pi }{8},\frac{5\pi }{8},\frac{7\pi }{8}$
• D. $0,\frac{\pi }{8},\frac{3\pi }{8},\frac{5\pi }{8},\frac{7\pi }{8},\pi$

Answer 1

The correct option is B $\frac{\pi }{8},\frac{\pi }{3},\frac{3\pi }{8},\frac{5\pi }{8},\frac{2\pi }{3},\frac{7\pi }{8}$

$\cos x\cos 2x\cos 3x=\frac{1}{4}$
$\Rightarrow 3{\cos }^{2}x-10{\cos }^{4}x+8{\cos }^{6}x=\frac{1}{4}$

Substitute $y={\cos }^{2}x$
$8y^3-10y^2+3y=\frac{1}{4}$
$\Rightarrow (4y-1)(8y^2-8y+1)=0$
$\Rightarrow y=\frac{1}{4}y=\frac{1}{2}\pm \frac{1}{2\sqrt{2}}$

$\Rightarrow {\cos }^{2}x=\frac{1}{4}$ or ${\cos }^{2}x=\frac{1}{2}\pm \frac{1}{2\sqrt{2}}$
$\Rightarrow x=\frac{\pi }{8},\frac{\pi }{3},\frac{3\pi }{8},\frac{5\pi }{8},\frac{2\pi }{3},\frac{7\pi }{8}$