Question

Find $\frac{dy}{dx}$, if $y=12(1-\cos t),x=10(t-\sin t),-\frac{\pi }{2}<t<\frac{\pi }{2}$

Answer 1

We have, $y=12(1-\cos t),x=10(t-\sin t)$
$\Rightarrow \frac{dx}{dt}=10(1-\cos t)$
& $\frac{dy}{dt}=12.[0-(-\sin t\left)\right]=12\sin t$
$\therefore \frac{dy}{dx}=\frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)}=\frac{12\sin t}{10(1-\cos t)}=\frac{12.2\sin \frac{t}{2}.\cos \frac{t}{2}}{10.2{\sin }^2\frac{t}{2}}=\frac{6}{5}\cot \frac{t}{2}$