Question

Find $\frac{dy}{dx}$, when

$x=a\left(\mathrm{\theta }+\sin \mathrm{\theta }\right)\mathrm{and}y=a\left(1-\cos \mathrm{\theta }\right)$

Answer 1

$\mathrm{We}\mathrm{have},x=a\left(\theta +\sin \theta \right)\mathrm{and}y=a\left(1-\cos \theta \right)$

$\Rightarrow \frac{dx}{d\theta }=a\left(1+\cos \theta \right)\mathrm{and}\frac{dy}{d\theta }=a\sin \theta$

$\therefore \frac{dy}{dx}=\frac{{\displaystyle \frac{dy}{d\theta }}}{{\displaystyle \frac{dx}{d\theta }}}=\frac{a\sin \theta }{a\left(1+\cos \theta \right)}=\frac{{\displaystyle 2\sin }{\displaystyle \frac{{\displaystyle \theta }}{2}}\cos {\displaystyle \frac{\theta }{2}}}{2{\cos }^2\theta }=\tan \frac{\theta }{2}$