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**Unformatted text preview: **hese representations by substituting the values for θ into r = 6 cos(2θ), once again, for illustration
purposes. (We feel most students would take this approach.) Again, in the interests of eﬃciency, we could ‘plug’
π
π
π
these values for θ into r = 3 (where there is no θ) and get the list of points: 3, π , 3, 23 , 3, 43 and 3, 53 .
3
While it is not true that 3, π represents the same point as −3, π , we still get the same set of solutions.
3
3
16
A quick sketch of r = 3 sin θ and r = 3 cos θ in the θr-plane will convince you that, viewed as functions of r,
2
2
these are two diﬀerent animals.
15 11.5 Graphs of Polar Equations 813
√ get only one intersection point which can be represented by 3 2 2 , π . We now investigate
2
other representations for the intersection points. Suppose P is an intersection point with
θ
a representation (r, θ) which satisﬁes r = 3 sin 2 and the same point P has a diﬀerent
θ
representation (r, θ + 2πk ) for some integer k which satisﬁes r = 3 cos 2 . Substituting
θ
into the latter, we get r = 3 cos...

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