Question

Find $\frac{dy}{dx},$ if $x=a(\theta +\sin \theta )$ and $y=a(1-\cos \theta )$.

Answer 1

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

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

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

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

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

$\frac{dy}{d\theta }=a\sin \theta$

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

$\frac{dy}{dx}=\frac{\sin \theta }{1+\cos \theta }$