Find the distance between the points with polar coordinates (4, π/3) and (8, 2π/3).

Solution:

Given polar coordinates (4, π/3) and (8, 2π/3).

We convert the polar coordinates (r, θ)into rectangular coordinates (x, y)

We know that (x, y) = (rsinθ, rcosθ)

(r, θ)₁ = (4, π/3)

(x, y)₁ = (4sin(π/3), 4cos(π/3))

Substituting the values, we get

(x, y)₁ = (4 ×√3/2  , 4 ×1/2)

(x, y)₁ = (2√3 ,2)

(r, θ)₂ = (8, 2π/3))

(x, y)₂ = (8sin(2π/3), 8cos(2π/3)))

Substituting the values, we get

(x, y)₂ = (8×√3/2 , 8×-1/2)

(x, y)₁ = (4√3, -4)

We know that distance between two cartesian points is given by

d = √{(x₂ - x₁)² +(y₂ - y₁)²}

d = √(4√3 -2√3)²+(-4 - 2)²

d =(2√3)²+(-6)²

d = 12 +36

d = 48

The distance between the points is 48.

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Find the distance between the points with polar coordinates (4, π/3) and (8, 2π/3).

Summary:

The distance between the points with polar coordinates (4, π/3) and (8, 2π/3) is 48