Question

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

Answer 1

It is given that ,y=12(1-cost),x=10(t-sint)
∴\(\frac{dy}{dx}\)=\(\frac{d}{dt}\)(10(t-sint))=10.\(\frac{d}{dt}\)(t-sint)=10(1-cost)
\(\frac{dy}{dt}\)=\(\frac{d}{dt}\)[12(1-cost)]=12.\(\frac{d}{dt}\)(1-cost)=12.[0-(-sint)]=12sint
∴\(\frac{dy}{dt}\)=\(\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) =\(\frac{12sin\,t}{10(1-cos\,t)}\)
=\(\frac{12.2sin\frac{t}{2}cos\frac{t}{2}}{10.2sin\frac{2t}{2}}\)=\(\frac{6}{5}cot\frac{t}{2}\)