Question

If $x=\cos t\mathrm{and}y=\sin t,$ prove that $\frac{dy}{dx}=\frac{1}{\sqrt{3}}\mathrm{at}t=\frac{2\pi }{3}$

Answer 1

$\mathrm{We}\mathrm{have},x=\cos t\mathrm{and}y=\sin t$

$$\begin{gathered} \Rightarrow \frac{dx}{dt}=\frac{d}{dt}\left(\cos t\right)\mathrm{and}\frac{dy}{dt}=\frac{d}{dt}\left(\sin t\right) \\ \Rightarrow \frac{dx}{dt}=-\sin t\mathrm{and}\frac{dy}{dt}=\cos t \\ \therefore \frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}=\frac{\cos t}{-\sin t}=-\cot t \\ \mathrm{Now},{\left(\frac{dy}{dx}\right)}_{t=\frac{2\mathrm{\pi }}{3}}=-\cot \left(\frac{2\mathrm{\pi }}{3}\right)=\frac{1}{\sqrt{3}} \end{gathered}$$