If x = a (1 – cos 3θ), y = a sin 3 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,
x = a (1 – cos 3θ) ……equation 1
y = a sin 3 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 [using chain rule]
Similarly,
……equation 4
[∵
Differentiating again w.r.t x :
…..equation 5
[ using chain rule and ]
From equation 3:
∴
Putting the value in equation 5 :
Put θ = π/6
∴