If x = a (1 – cos θ), y =a (θ + sin θ), prove that
Idea of parametric form of differentiation:
If y = f (θ) and x = g(θ) i.e. y is a function of θ and x is also some other function of θ.
Then dy/dθ = f’(θ) and dx/dθ = g’(θ)
We can write :
Given,
y = a (θ + sin θ) ……equation 1
x = a (1– cos θ) ……equation 2
to prove : .
We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative.
Let’s find
As
So, lets first find dy/dx using parametric form and differentiate it again.
…..equation 3
Similarly,
……equation 4
[∵
…..equation 5
Differentiating again w.r.t x :
Using product rule and chain rule of differentiation together:
[using equation 4]
As we have to find
∴ put θ = π/2 in above equation:
=