Question

if x= ϴsin​​​ ϴ y = 1-cos ϴ find dy/dx at ϴ = π/2

Answer 1

Dear student

$$\begin{gathered} Given: \\ x=\theta \sin \theta \\ Differentiatewrt\theta \\ \frac{dx}{d\theta }=\theta \cos \theta +\sin \theta \\ y=1-\cos \theta \\ Differentiatewrt\theta \\ \frac{dy}{d\theta }=\sin \theta \\ Now, \\ \frac{dy}{dx}=\frac{\frac{dx}{d\theta }}{\frac{dy}{d\theta }}=\frac{\theta \cos \theta +\sin \theta }{\sin \theta } \\ {\left(\frac{dy}{dx}\right)}_{\theta =\frac{\mathrm{\pi }}{2}}=\frac{\frac{\mathrm{\pi }}{2}\cos \left(\frac{\mathrm{\pi }}{2}\right)+\sin \left(\frac{\mathrm{\pi }}{2}\right)}{\sin \frac{\mathrm{\pi }}{2}} \\ =\frac{0+1}{1} \\ =1 \end{gathered}$$
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