Question

If $cosθ+cos2θ+cos3θ=0$, then the general value of $θ$ is :

• A. $θ=2mÏ€Â\pm 2Ï€/3$
• B. $θ=2mÏ€Â\pm 2Ï€/4$
• C. $θ=mÏ€+{\left(−1\right)}^{n}2Ï€/3$
• D. $θ=mÏ€+{\left(−1\right)}^nÏ€/3$

Answer 1

Answer: (A)

Given $cosθ+cos2θ+cos3θ=0$
$⇒\left(cos3θ+cosθ\right)+cos2θ=0$
$⇒2cos2θ.cosθ+cos2θ=0$
$⇒cos2θ.\left(2cosθ+1\right)=0$
we have, $cosθ=cosÎ\pm ⇒θ=2nÏ€Â\pm Î\pm$
$∴$ For general value of $θ,cos2θ=0$
$⇒cos2θ=cos\frac{Ï€}{2}⇒2θ=2mÏ€Â\pm \frac{Ï€}{2}$
$⇒θ=mÏ€Â\pm \frac{Ï€}{4}$ or $2cosθ+1=0;$
$⇒cosθ=\frac{−1}{2}⇒cosθ=cos\frac{2π}{3}$
So, $θ=2mÂ\pm \frac{2Ï€}{3}$