Find all polar coordinates of point p where p = ordered pair 5 comma pi divided by 3.

Solution:

The polar coordinates of any point can be written as

(r,θ) = (r, θ+2n𝜋) (when r is positive)

(r,θ) = (-r, θ+(2n+1)𝜋) (when r is negative)

Where n is an integer.

Given the point is (5, 𝜋/3)

When r is positive, the polar coordinate is (5, 𝜋/3 + 2n𝜋)

When r is negative, the polar coordinate is (-5, (2n+1)𝜋)

Therefore, all polar coordinates of a given point are (5, 𝜋/3 + 2n𝜋) and (-5, (2n+1)𝜋).

Find all polar coordinates of point p where p = ordered pair 5 comma pi divided by 3.

Summary:

All polar coordinates of point p where p = ordered pair 5 comma pi divided by 3 is (5, 𝜋/3 + 2n𝜋) and (-5, (2n+1)𝜋).