Q. If $x = a(\theta - \sin{\theta),\ y = a\left( 1 - \cos\theta \right),then\frac{dy}{dx}}\$is equal to
A) $\cot\frac{\theta}{2}$
B) $\tan\frac{\theta}{2}$
C) $\frac{1}{2}{cosec}^{2}\frac{\theta}{2}\$
D) $- \frac{1}{2}{cosec}^{2}\frac{\theta}{2}$
Explanation
(a)
Given, $x = a(\theta - \sin\theta$) and $y = a(1 - \cos{\theta)}$
$\ \frac{dx}{d\theta}\ = \ a\left( 1 - \cos\theta \right)\ \$and $\frac{dy}{d\theta}\ = \ {a\ sin}\theta$
$\therefore\frac{dy}{dx} = \frac{a\sin\theta}{a(1 - \cos{\theta)}}$
$= \frac{2{sin\ }\frac{\theta}{2}\cos{\ \frac{\theta}{2}}}{2{\sin^{2}\ }\frac{\theta}{2}} = \cot\frac{\theta}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.