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...
View
Full Document
 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

Click to edit the document details