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Unformatted text preview: ) represent the pole regardless of the value of θ.
The key to understanding this result, and indeed the whole polar coordinate system, is to keep in
mind that (r, θ) means (directed distance from pole, angle of rotation). If r = 0, then no matter
how much rotation is performed, the point never leaves the pole. Thus (0, θ) is the pole for all
values of θ. Now let’s assume that neither r nor r is zero. If (r, θ) and (r , θ ) determine the same 11.4 Polar Coordinates 787 point P then the (non-zero) distance from P to the pole in each case must be the same. Since this
distance is controlled by the ﬁrst coordinate, we have that either r = r or r = −r. If r = r, then
when plotting (r, θ) and (r , θ ), the angles θ and θ have the same initial side. Hence, if (r, θ) and
(r , θ ) determine the same point, we must have that θ is coterminal with θ. We know that this
means θ = θ + 2πk for some integer k , as required. If, on the other hand, r = −r, then when
plotting (r, θ) and (r , θ ), the initial side of θ i...
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