Question

Find $\frac{dy}{dx}$ in

$x=a(\theta -\sin \theta )$
$y=b(1-\cos \theta )$

Answer 1

$x=a(\theta -\sin \theta )$

Differentiate both sides w.r.t. $\theta$

$\frac{dx}{d\theta }=a(1-\cos \theta )$

$y=b(1-\cos \theta )$

Differentiate both sides w.r.t. $\theta$

$\frac{dy}{d\theta }=b\left(\sin \theta \right)$

Now,

$\frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$

$=\frac{b\sin \theta }{a(1-\cos \theta )}$

$=\frac{2b\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{2a{\sin }^2\frac{\theta }{2}}$

$=\frac{b}{a}\cot \frac{\theta }{2}$