Question

Solve $2{\cos }^2\theta +\cos \theta -1=0$

• A. $2n\pi \forall n\in N$
• B. $2n\pi \mp \frac{\pi }{3}\forall n\in N$
• C. $2n\pi \mp \frac{\pi }{6}\forall n\in N$
• D. $(2n+1)\frac{\pi }{6}\forall n\in N$

Answer 1

The correct option is B

$2n\pi \mp \frac{\pi }{3}\forall n\in N$

We can see that the given expression is a quadratic in cos. We can find its roots and find the general solutions easily.

$\Rightarrow \cos \theta =\frac{-1\mp \sqrt{1+8}}{4}$
$\cos \theta =\frac{1}{2}$ or $\cos \theta =-1$

We know that the general solution forumla for the equation
$\cos x=\cos \alpha$
is $x=2n\pi \pm \alpha$
$\Rightarrow \theta =2n\pi \pm \frac{\pi }{3}$ or $\pi =(2n\pm 1)\pi$