Q. 174.6( 7 Votes )
If x = cos θ, y = sin3θ. Prove that
The 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 :
y = sin3θ ……equation 1
x = cos θ ……equation 2
We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative.
So, lets first find dy/dx using parametric form and differentiate it again.
Applying chain rule to differentiate sin3θ :
Again differentiating w.r.t x:
Applying product rule and chain rule to differentiate:
[using equation 3 to put the value of dθ/dx]
Multiplying y both sides to approach towards the expression we want to prove-
[from equation 1, substituting for y]
Adding equation 5 after squaring it:
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