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

1 Answer
Jun 5, 2018

Answer:

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").