Q. 174.6( 7 Votes )

# If x = cos θ, y = sin^{3}θ. Prove that

Answer :

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 :

Given,

y = sin^{3}θ ……equation 1

x = 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

Applying chain rule to differentiate sin^{3}θ :

…………..equation 4

………..equation 5

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:

Rate this question :

If y = 3 cos(log x) + 4 sin(log x), show that

Mathematics - Board Papers

If x = t^{2}, y = t^{3}, then is

If x = a cos θ + b sin θ, y = a sin θ – b cos θ, show that

Mathematics - Board PapersIf prove that

Mathematics - Exemplar

If y =(tan^{–1}x)^{2}, show that

If y = (tan ^{- 1} x)^{2}, show that (x^{2} + 1)^{2}

Differentiate

**OR**

If x = a(θ – sin θ), y = a(1 + cosθ), find

Mathematics - Board PapersFind if (x^{2} + y^{2})^{2} = xy.

**OR**

If y = 3 cos(log x) + 4 sin(log x), then show that

Mathematics - Board Papers