Question

Find $\frac{dy}{dx},$ when
$x=a(1-\cos \theta )$ and $y=a(\theta +\sin \theta )$ at $\theta =\frac{\pi }{2}$

Answer 1

We have,

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

Differentiating w.r.t. $\theta$, we get,

$\Rightarrow \frac{dx}{d\theta }=a\sin \theta ,\frac{dy}{d\theta }=a(1+\cos \theta )$

$\therefore \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=\frac{a(1+\cos \theta )}{a\sin \theta }=\frac{2{\cos }^2\frac{\theta }{2}}{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}=\cot \frac{\theta }{2}$

$\Rightarrow {\left(\frac{dy}{dx}\right)}_{\theta =\frac{\pi }{2}}=\cot \frac{\pi }{4}=1.$