Question

If $x=a(\theta +\sin \theta )$ and $y=a(1-\cos \theta )$, find $dy/dx$.

Answer 1

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

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

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

$\Rightarrow \frac{dy}{d\theta }=a(0-(-\sin \theta \left)\right)=a\sin \theta$ (2)

Dividing (2) and (1) $\frac{dy/d\theta }{dx/d\theta }=\frac{dy}{dx}=\frac{\sin \theta }{1+\cos \theta }=tan\left(\theta /2\right)$