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Unformatted text preview: s rotated π radians away from the initial side of
θ. In this case, θ must be coterminal with π + θ. Hence, θ = π + θ + 2πk which we rewrite as
θ = θ + (2k + 1)π for some integer k . Conversely, if r = r and θ = θ + 2πk for some integer k , then
the points P (r, θ) and P (r , θ ) lie the same (directed) distance from the pole on the terminal sides
of coterminal angles, and hence are the same point. Now suppose r = −r and θ = θ + (2k + 1)π
for some integer k . To plot P , we ﬁrst move a directed distance r from the pole; to plot P , our
ﬁrst step is to move the same distance from the pole as P , but in the opposite direction. At this
intermediate stage, we have two points equidistant from the pole rotated exactly π radians apart.
Since θ = θ + (2k + 1)π = (θ + π ) + 2πk for some integer k , we see that θ is coterminal to (θ + π )
and it is this extra π radians of rotation which aligns the points P and P .
Next, we marry the polar coordinate system with the Cartesian (rectangu...
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