Question

Find the value of `dy/dx` at `θ = π/3`, if x = cos θ – cos 2θ, y = sin θ – sin 2θ.

Answer 1

Given, x = cos θ – 2 cos 2θ

y = sin θ – sin 2θ

Differentiate x = cos θ – 2 cos 2θ w.r.t θ

`dx/(dθ) = - sin θ + 2 sin 2θ`   ...(i)

Differentiate y = sin θ – sin 2θ w.r.t θ

`dy/(dθ) = cos θ - cos 2θ`   ...(ii)

On dividing equation (ii) by equation (i)

`((dy)/(dθ))/((dx)/(dθ)) = (cos θ - 2 cos 2θ)/(-sin θ + 2 sin 2θ)`

`dy/dx = (cos θ - 2 cos 2θ)/(2 sin 2θ - sin θ)`

`dy/dx]_(θ = π/3) = (cos(π/3) - 2 cos 2(π/3))/(2 sin 2(π/3) - sin(π/3))`

= `((1/2) - 2((-1)/2))/(2(sqrt(3)/2) - (sqrt(3)/2)`

= `(1/2 + 1)/(sqrt(3)/2)`

= `(3/2)/(sqrt(3)/2)`

= `sqrt(3)`