Question

cos theta cos 2theta cos 3 theta = 1/4​

Answer 1

Answer:

Here is your answer

Step-by-step explanation:

Answer 2

Answer:

The value of $\theta =(n \pm1)\dfrac{\pi }{3}~ \mbox{or}~(2n \pm1)\dfrac{\pi }{2}~ \mbox{for}~n=\pm0,\pm1,\pm2,...$

Step-by-step explanation:

Given the trigonometric equation,  

$cos \theta ~cos 2\theta~ cos 3\theta= \dfrac{1}{4}$

$\implies 4cos \theta ~cos 2\theta~ cos 3\theta= 1$

$\implies 2cos \theta ~ cos 3\theta.2~cos 2\theta= 1$

Using the trigonometric relation, $2cosAcosB= cos( A +B)+cos( A -B)$, we get

$\big(cos( \theta +3\theta)+cos( \theta -3\theta)\big).2cos2\theta= 1$

$\implies \big(cos( 4\theta)+cos( -2\theta)\big).2cos2\theta= 1$

$\implies 2cos2\theta cos 4\theta+2cos2\theta cos 2\theta\big=1$

$\implies 2cos2\theta cos 4\theta+2cos^22\theta -1=0$

Using the trigonometric relation, $2cos^2\theta -1=cos2\theta \implies 2cos^22\theta -1=cos4\theta$, we get

$2cos2\theta cos 4\theta+cos4\theta=0$

$\implies cos4\theta (2cos 2\theta+1)=0$

$\implies 2cos 2\theta+1=0$ or $cos4\theta =0$

From $cos4\theta =0$ we get

$\implies cos4\theta =cos\Big(2n\pi \pm\dfrac{\pi }{2}\Big)$

$\implies4\theta =(2n \pm1)\dfrac{\pi }{2} \implies\theta =(2n \pm1)\dfrac{\pi }{8}$

Similarly from $2cos 2\theta+1=0$

$\implies cos 2\theta=-\dfrac{1}{2}\implies cos2\theta =cos\Big(2n\pi \pm\dfrac{2\pi }{3}\Big)$

$\implies 2\theta =2n\pi \pm\dfrac{2\pi }{3}\implies \theta =(n \pm1)\dfrac{\pi }{3}$

Therefore, the value of $\theta =(n \pm1)\dfrac{\pi }{3}~ \mbox{or}~(2n \pm1)\dfrac{\pi }{2}~ \mbox{for}~n=\pm0,\pm1,\pm2,...$