Writing Equivalent Polar Coordinates :
Let P have a polar coordinates (r, θ). Any other polar coordinate P must be in the following form.
(r, θ + 2nπ) or (-r, θ + (2n + 1)π)
Where n is any integer. In particular, the pole has polar coordinates (0, θ), where θ is any angle.
Example 1 :
Sketch (5, 3π/4) and identify 3 other polar coordinates on the interval (-2π, 2π) that represent the same point.
First let us plot the point (5, 3π/4) in the polar grid.
From this position, we can move either clock wise or anti clock wise.
r > 0, θ = 360 - 135 ==> 225 ==> 5π/4
The another point is (5, 5π/4).
If r < 0, then θ = π + 3π/4 ==> 7π/4
The point on the terminal side is (-3, 7π/4). So the another point is (-3, π/4).
Example 2 :
Find all polar coordinates of the point (4, 4√3) that describes this point given that 0 < θ < 2π
x = 4 and y = 4√3
r cosθ = 4 and r sin θ = 4√3
x2 + y2 = r2
42 + (4√3)2 = r2
r2 = 16 + 16(3)
r2 = 64
r = ± 8
If r = 8 > 0, then θ = cos-1(4/8)
θ = cos-1(1/2)
θ = π/3
(8, π/3) and (8, 5π/3).
If r = -8 < 0, then θ = cos-1(-4/8)
θ = cos-1(-1/2)
θ = 2π/3
(-8, 2π/3) and (8, 4π/3).
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