The polar coordinates of point P are given. Find all of its polar coordinates.
-$$\mathrm { P } = ( 9 , \pi / 3 )$$
A) $( 9 , \pi / 3 + 2 \mathrm { n } \pi )$ or $( - 9 , \pi / 3 + 2 \mathrm { n } \pi )$
B) $( 9 , \pi / 3 + ( 2 n + 1 ) \pi )$ or $( - 9 , \pi / 3 + 2 n \pi )$
C) $( 9 , \pi / 3 + 2 n \pi )$ or $( - 9 , \pi / 3 + ( 2 n + 1 ) \pi )$
D) $( 9 , \pi / 3 + n \pi )$ or $( - 9 , \pi / 3 + n \pi )$

Question

Quizplus · Accepted Answer

In polar coordinates, a point can be represented in multiple forms. For a given radius $r$ and angle $\theta$, the point can also be represented as $(r, \theta + 2n\pi)$ for any integer $n$, because adding $2\pi$ radians (or a full circle) to the angle results in the same direction. Additionally, the point $(-r, \theta)$ is the same as $(r, \theta + \pi)$ because flipping the sign of the radius is equivalent to moving the point $180^\circ$ (or $\pi$ radians) around the circle. Therefore, the correct representations for all polar coordinates of point $P$ are $(9, \pi/3 + 2n\pi)$ and $(-9, \pi/3 + (2n+1)\pi)$, where the latter accounts for the angle adjustment needed when the radius is negative.

The Polar Coordinates of Point P Are Given P=(9,π/3)\mathrm { P } = ( 9 , \pi / 3 )P=(9,π/3)

The Polar Coordinates of Point P Are Given $\mathrm { P } = ( 9 , \pi / 3 )$