Question

Find all points (if any) of horizontal and vertical tangency to the portion of the curve. Involute of a circle: x = cos θ + θ sin θ y = sinθ - θ cos θ

Answer 1

Differentiating x w.r.t o̶ dx/do̶=-sin o̶+sin o̶+o̶ cos o̶ dx/do̶=o̶ cos o̶ y w.r.t. o̶ dy/do̶= cos o̶-cos o̶+ o̶ sin o̶ dy/do̶=o̶ sin o̶ For horizontal tangents dy/do̶=0 o̶ sin o̶ = 0 o̶= 0,± π,±2π,±3π,±4π,... The values of coordinates cos )are, x=cos o̶+o̶ sin o̶ = ±1 y=sin o̶-cos o̶=±π,±2π,±3π,±4π,... For vertical tangents, dx/do̶=0 o̶ cos o̶=0 O̶=±π/2, ±3π/2 ,±5π/2 ,±7π/2,... The values of coordinates are, x=cos o̶+sin o̶=±π/2,±3π/2,±5π/2,±7π/2,... y=sin o̶ - o̶ cos o̶=±1 For horizontal tangents (1,0),(-1, π),(1,-2π ) For vertical tangents (π/2,1),(-3π/2,-1),(5π/2,1)