Parametric equations and differentiation?

Question

Any help with this maths question would be appreciated:

A curve C has parametric equations x=cos(theta) and
y=cos(3theta)

Where 0 less than equal to theta more than equal to pi

The first part is to show that y=4(x^3) - 3x which i've done

Then it wants to show dy/dx in terms of x, which is 12(x^2) -3

And then find dy/dx in terms of theta (which i think i can do)

And then show that

sin (3theta) =sin theta (3-4sin^2(theta)) - WHich i cant do

Can you maybe explain the answer as well????

Answer 1

x = cos(θ)
y = cos(3θ)
and you've already worked out that y = 4x³ - 3x

So dy/dx = 12x² - 3
So dy/dx = 12cos²(θ) - 3

Now use y = cos(3θ) and take a derivative of both sides with respect to θ (our motivation in doing this is that the final answer has a sin(3θ) and differentiating cos(3θ) is a good way to get it):
dy/dθ = -3sin(3θ) = dy/dx * dx/dθ = 3(4cos²(θ)-1) * (-sin(θ))

Divide through by -3:
sin(3θ) = (4cos²(θ)-1) * sin(θ)
= (3 - 4sin²(θ)) * sin(θ)