Question

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

Answer 1

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 `(1.5,180^o)` is equivalent also to `(1.5,540^o)`, `(1.5,900^o)`, 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:
`(1.5,-180^o)`, `(1.5,-540^o)`, `(1.5,-900^o)`, 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 `(-1.5,0^o)`.

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

We can describe these families of alternates more compactly:
`(1.5, (180+360n)^o)`, `(-1.5,360n)^o`, `AA n in ZZ` (which bit of mathematical symbolism reads in English "for all numbers n that are integers").