**Unformatted text preview: **ntation for P with r = −3, we may choose any angle coterminal with
π
π
− π . We choose θ = 74 for our ﬁnal answer −3, 74 .
4 P 3, 3π
4 P −3,
θ= 7π
4 3π
4 Pole θ= 7π
4 Pole Now that we have had some practice with plotting points in polar coordinates, it should come as
no surprise that any given point expressed in polar coordinates has inﬁnitely many other representations in polar coordinates. The following result characterizes when two sets of polar coordinates
determine the same point in the plane. It could be considered as a deﬁnition or a theorem, depending on your point of view. We state it as a property of the polar coordinate system.
Equivalent Representations of Points in Polar Coordinates
Suppose (r, θ) and (r , θ ) are polar coordinates where r = 0, r = 0 and the angles are measured
in radians. Then (r, θ) and (r , θ ) determine the same point P if and only if one of the following
is true:
• r = r and θ = θ + 2πk for some integer k
• r = −r and θ = θ + (2k + 1)π for some integer k
All polar coordinates of the form (0, ...

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