Question

If x = a(cos 2θ + 2θ sin 2θ) and y = a(sin 2θ − 2θ cos 2θ) ,
find d²y/dx² at θ = π/8 .

Answer 1

Explanation:

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