Question

If $x=a\left(\mathrm{\theta }+\sin \mathrm{\theta }\right),y=a\left(1+\cos \mathrm{\theta }\right),\mathrm{find}\frac{dy}{dx}$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},x=a\left(\theta +\sin \theta \right)\mathrm{and}y=a\left(1+\cos \theta \right) \\ \Rightarrow \frac{dx}{d\theta }=a\left\{\frac{d}{d\theta }\left(\theta \right)+\frac{d}{d\theta }\left(\sin \theta \right)\right\}\mathrm{and}\frac{dy}{d\theta }=a\left(0-\sin \theta \right) \\ \Rightarrow \frac{dx}{d\theta }=a\left(1+\cos \theta \right)\mathrm{and}\frac{dy}{d\theta }=-a\sin \theta \\ \therefore \frac{dy}{dx}=\frac{{\displaystyle \frac{dy}{d\theta }}}{{\displaystyle \frac{dx}{d\theta }}}=\frac{-a\sin \theta }{a\left(1+\cos \theta \right)}=\frac{-2{\displaystyle \sin }{\displaystyle \frac{{\displaystyle \theta }}{2}}{\displaystyle \cos \frac{\theta }{2}}}{2{\cos }^2{\displaystyle \frac{\theta }{2}}}=-\tan \frac{\theta }{2} \end{gathered}$$