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Unformatted text preview: nitely many representations. As a result, if a point P is on the graph of two diﬀerent polar
equations, it is entirely possible that the representation P (r, θ) which satisﬁes one of the equations
does not satisfy the other equation. Here, more than ever, we need to rely on the Geometry as
much as the Algebra to ﬁnd our solutions.
Example 11.5.3. Find the points of intersection of the graphs of the following polar equations.
1. r = 2 sin(θ) and r = 2 − 2 sin(θ) 3. r = 3 and r = 6 cos(2θ) 2. r = 2 and r = 3 cos(θ) 4. r = 3 sin θ
2 and r = 3 cos θ
1. Following the procedure in Example 11.5.2, we graph r = 2 sin(θ) and ﬁnd it to be a circle
centered at the point with rectangular coordinates (0, 1) with a radius of 1. The graph of
r = 2 − 2 sin(θ) is a special kind of lima¸on called a ‘cardioid.’10
2 −2 2 x −4 r = 2 − 2 sin(θ) and r = 2 sin(θ )
It appears as if there are three intersection points: one in the ﬁrst quadrant, one in the second
quadrant, and the origin. Our next task...
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