Question

x=a(2theta-sin2theta) and y=a(1-cos2theta) find dy/dx when theta = π/3

Answer 1

x = a(cos2θ - sin2θ)
difference x with respect to θ
dx/dθ = a[d(cos2θ)/dθ - d(sin2θ)/dθ]
dx/dθ = a[-2sin2θ - 2cos2θ] = -2a(sin2θ + cos2θ) ----(1)

Similarly, y = a(1 - cos2θ)
differentiate with respect to θ
dy/dθ = a[0 - d(cos2θ)/dθ]
dy/dθ = 2asin2θ -----(2)

Now, dividing equation (2) ÷ (1),
{dy/dθ}/{dx/dθ} = -2a(sin2θ + cos2θ)/2asin2θ = -(sin2θ + cos2θ)/sin2θ
dy/dx = -(sin2θ + cos2θ)/sin2θ
at θ = π/3 , dy/dx = -(√3/2 - 1/2)/√3/2 = -(√3 - 1)/√3

Hence, dy/dx at π/3 is -(√3 - 1)/√3

Answer 2

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