# How do you find four other pairs of polar coordinates for the point T(1.5, 180^o)?

Jun 5, 2018

Consider moving the angle around the axis multiple times. Also consider the effect of moving radially backwards instead of forwards.

#### Explanation:

Think of how the polar coordinate system works - you have a radius $R$ from the origin ("pole"), and it sweeps around the origin from the chosen "polar axis", the line on which the angle $\theta$ is zero.

So there's immediately one way in which a point can have more than one description in polar coordinates - by going around the angle more than once. So adding ${360}^{o}$ (or another multiple of ${360}^{0}$) to any polar coordinate will bring you back to the same place. So the given example of $\left(1.5 , {180}^{o}\right)$ is equivalent also to $\left(1.5 , {540}^{o}\right)$, $\left(1.5 , {900}^{o}\right)$, etc.

Don't forget that we also go back around the angle full turns in the other direction, obtaining also an infinite number of negative angles that are equivalent to the point $T$ by subtracting off multiples of ${360}^{o}$ instead:
$\left(1.5 , - {180}^{o}\right)$, $\left(1.5 , - {540}^{o}\right)$, $\left(1.5 , - {900}^{o}\right)$, etc.

The final way in which we can find alternate descriptions of a polar point is to consider making the radial coordinate $R$ negative. Moving in some angular direction a distance $R$ is equivalent to moving in the opposite angular direction a distance $- R$. The opposite direction is ${180}^{o}$ away - so the simplest angle for the opposite direction is ${0}^{o}$. Thus T is also $\left(- 1.5 , {0}^{o}\right)$.

As before, we can find an infinite number of extra angular turns in both positive and negative directions. These give us alternate coordinate descriptions $\left(- 1.5 , {360}^{o}\right)$, $\left(- 1.5 , {720}^{o}\right)$, $\left(1.5 , {1080}^{o}\right)$, etc. and $\left(- 1.5 , - {360}^{o}\right)$, $\left(- 1.5 , - {720}^{o}\right)$, $\left(1.5 , - {1080}^{o}\right)$, etc.

We can describe these families of alternates more compactly:
$\left(1.5 , {\left(180 + 360 n\right)}^{o}\right)$, ${\left(- 1.5 , 360 n\right)}^{o}$, $\forall n \in \mathbb{Z}$ (which bit of mathematical symbolism reads in English "for all numbers n that are integers").