Find a set of polar coordinates for the point for which the rectangular coordinates are given. (3, -3 roots of 3  )?

Converting from rectangular to polar we have:

(x, y) => (r, θ)

Now, r² = 3² + (-3√3)² => 9 + 27 = 36

Hence, r = 6

Now, tanθ = (-3√3)/3 => -√3

so, θ = tan⁻¹(-√3) => -60°.....i.e. 60° anticlockwise

Therefore, θ = 300° anticlockwise

so, (3, -3√3) => (6, 300°)

Checking gives:

x = 6cos300° => 3

y = 6sin300° => -3√3

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https://www.mathsisfun.com/polar-cartesian-coordinates.html

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Rectangular coordinates (3, -3√3) to (r, θ).
As x > 0 and y < 0, θ is in the 4th quadrant.

r = √[3² + (-3√3)²]
r = √(36)
r = 6

θ = tan⁻¹(-3√3/3)
θ = tan⁻¹(-√3)
θ = 360° - 60°
θ = 300°

Polar coordinates = (6, 300°)

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