Question

Find $\frac{dy}{dx}$, if $x=2\cos \theta -\cos 2\theta$ and $y=2\sin \theta -\sin 2\theta$.

• A. $\tan \frac{3\theta }{2}$
• B. $-\tan \frac{3\theta }{2}$
• C. $\cot \frac{3\theta }{2}$
• D. $-\cot \frac{3\theta }{2}$

Answer 1

The correct option is A $\tan \frac{3\theta }{2}$

Given, $x=2\cos \theta -\cos 2\theta$ and $y=2\sin \theta -\sin 2\theta$
$\frac{dx}{d\theta }=-2\sin \theta +2\sin 2\theta$
and $\frac{dy}{d\theta }=2\cos \theta -2\cos 2\theta$
$\therefore \frac{dy}{dx}=\frac{2\cos \theta -2\cos 2\theta }{-2\sin \theta +2\sin 2\theta }$
$=\frac{\cos \theta -\cos 2\theta }{\sin 2\theta -\sin \theta }$
$=\frac{2\sin \left(\frac{\theta +2\theta }{2}\right)\sin \left(\frac{2\theta -\theta }{2}\right)}{2\cos \left(\frac{\theta +2\theta }{2}\right)\sin \left(\frac{2\theta -\theta }{2}\right)}$
$=\tan \frac{3\theta }{2}$