Find the distance between the points with polar coordinates 4√3 and 8√2

To find the distance between two points given in polar coordinates, we can use the formula:

d = √[(r₁ * cos(θ₁) – r₂ * cos(θ₂))² + (r₁ * sin(θ₁) – r₂ * sin(θ₂))²]

Here, the polar coordinates are given as:

• Point 1: (r₁, θ₁) = (4√3, π/3)
• Point 2: (r₂, θ₂) = (8, 2π/3)

In order to apply the formula, first we need to convert the polar coordinates to Cartesian coordinates:

• For Point 1:
  • x₁ = r₁ * cos(θ₁) = 4√3 * cos(π/3) = 4√3 * 1/2 = 2√3
  • y₁ = r₁ * sin(θ₁) = 4√3 * sin(π/3) = 4√3 * (√3/2) = 6
• For Point 2:
  • x₂ = r₂ * cos(θ₂) = 8 * cos(2π/3) = 8 * (-1/2) = -4
  • y₂ = r₂ * sin(θ₂) = 8 * sin(2π/3) = 8 * (√3/2) = 4√3

Next, we can plug these coordinates into the distance formula:

d = √[(2√3 – (-4))² + (6 – 4√3)²]

d = √[(2√3 + 4)² + (6 – 4√3)²]

Calculating these values:

• (2√3 + 4)² = (2√3)² + 2 * 2√3 * 4 + 4² = 12 + 16√3 + 16 = 28 + 16√3
• (6 – 4√3)² = 6² – 2 * 6 * 4√3 + (4√3)² = 36 – 48√3 + 48 = 84 – 48√3

Now substituting back into the distance formula gives:
d = √[(28 + 16√3) + (84 – 48√3)]
= √[112 – 32√3]

Hence, the distance between the two points is d = √[112 – 32√3].