Question

How do you find all polar coordinates of point P where $P = \left( {9,\dfrac{{2\pi }}{3}} \right)$?

Answer 1

In this question we are asked to find all polar coordinates of the given point, this can be done by using the polar coordinate formula which is given by, when $r$ is positive the point will be $\left( {r,\theta + 2n\pi } \right)$and when $r$ is negative the point will be$\left( { - r,\theta + \left( {2n + 1} \right)\pi } \right)$, where n is represented as an integer. By substituting the values in the formula we will get the required polar coordinates of the given point.

Complete step by step solution:
Given point is $P = \left( {9,\dfrac{{2\pi }}{3}} \right)$,
We know that we can derive any number of polar coordinates for one coordinate point by using the formula, when $r$ is positive the point will be $\left( {r,\theta + 2n\pi } \right)$ and when $r$ is negative the point will be $\left( { - r,\theta + \left( {2n + 1} \right)\pi } \right)$, where n is represented as an integer.
Now given coordinates are, $P = \left( {9,\dfrac{{2\pi }}{3}} \right)$,
Here $r = 9$ and $\theta = \dfrac{{2\pi }}{3}$,
Now substituting the values in the formula, we get,
For positive $r$, the polar coordinate can be written as,
$P = \left( {9,\dfrac{{2\pi }}{3}} \right) = \left( {9,\dfrac{{2\pi }}{3} + 2n\pi } \right)$, where n is an integer,
For negative $r$, the polar coordinate can be written as,
$P = \left( {9,\dfrac{{2\pi }}{3}} \right) = \left( { - 9,\dfrac{{2\pi }}{3} + \left( {2n + 1} \right)\pi } \right)$, where n is an integer,
So, all polar coordinates are, $\left( {9,\dfrac{{2\pi }}{3} + 2n\pi } \right)$and $\left( { - 9,\dfrac{{2\pi }}{3} + \left( {2n + 1} \right)\pi } \right)$.
Final Answer:
$\therefore$ All polar coordinates of point P where $P = \left( {9,\dfrac{{2\pi }}{3}} \right)$ are given as $\left( {9,\dfrac{{2\pi }}{3} + 2n\pi } \right)$ and $\left( { - 9,\dfrac{{2\pi }}{3} + \left( {2n + 1} \right)\pi } \right)$.

Note:
Polar coordinates are based on angles, unlike Cartesian coordinates, polar coordinates have many different ordered pairs. Because infinitely many values of theta have the same angle in standard position, an infinite number of coordinate pairs describe the same point. Also, a positive and a negative coterminal angle can describe the same point for the same radius, and because the radius can be either positive or negative, you can express the point with polar coordinates in many ways.