Question

If $\frac{1}{6}\sin \theta$, $\cos \theta$ and $\tan \theta$ are in $GP$ then the general value of $\theta$ is

• A. $2n\pi \pm \frac{\pi }{3},n\in 1$
• B. $2n\pi \pm \frac{\pi }{6},n\in 1$
• C. $n\pi +{(-1)}^n\frac{\pi }{3},n\in 1$
• D. $n\pi \pm \frac{\pi }{3},n\in 1$

Answer 1

The correct option is A $2n\pi \pm \frac{\pi }{3},n\in 1$

They are in GP, so
$\frac{\sin \theta \tan \theta }{6}={\cos }^2\theta$

$\Rightarrow {\sin }^2\theta =6{\cos }^3\theta$

$\Rightarrow 1-{\cos }^2\theta =6{\cos }^3\theta$

$\Rightarrow \cos \theta =\frac{1}{2}or\frac{-1-i\sqrt{2}}{3}or\frac{-1+i\sqrt{2}}{3}$

$\Rightarrow \cos \theta =\cos \frac{\pi }{3}$

$\Rightarrow \theta =2n\pi \pm \frac{\pi }{3}$