Question

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

$x=a\left(1-\cos \mathrm{\theta }\right)\mathrm{and}y=a\left(\mathrm{\theta }+\sin \mathrm{\theta }\right)at\mathrm{\theta }=\frac{\pi }{2}$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},x=a\left(1-\cos \theta \right)andy=a\left(\theta +\sin \theta \right) \\ \therefore \frac{dx}{d\theta }=\frac{d}{d\theta }\left[a\left(1-\cos \theta \right)\right]=a\left(\sin \theta \right) \\ \mathrm{and} \\ \frac{dy}{d\theta }=\frac{d}{d\theta }\left[a\left(\theta +\sin \theta \right)\right]=a\left(1+\cos \theta \right) \\ \therefore {\left[\frac{dy}{dx}\right]}_{\theta =\frac{\mathrm{\pi }}{2}}={\left[\frac{{\displaystyle \frac{dy}{d\theta }}}{{\displaystyle \frac{dx}{d\theta }}}\right]}_{\theta =\frac{\mathrm{\pi }}{2}}={\left[\frac{a\left(1+\cos \theta \right)}{a\left(\sin \theta \right)}\right]}_{\theta =\frac{\mathrm{\pi }}{2}}=\frac{a\left(1+0\right)}{a}=1 \end{gathered}$$