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**Unformatted text preview: **r = 3 cos 2 . Using the techniques
presented in Example 11.5.2, we ﬁnd that we need to plot both functions as θ ranges from
0 to 4π to obtain the complete graph. To our surprise and/or delight, it appears as if these
two equations describe the same curve!
y
3 −3 3 x −3
θ
r = 3 sin 2 and r = 3 cos θ
2
appear to determine the same curve in the xy -plane To verify this incredible claim,16 we need to show that, in fact, the graphs of these two
equations intersect at all points on the plane. Suppose P has a representation (r, θ) which
θ
θ
satisﬁes both r = 3 sin 2 and r = 3 cos 2 . Equating these two expressions for r gives
θ
θ
the equation 3 sin 2 = 3 cos 2 . While normally we discourage dividing by a variable
θ
expression (in case it could be 0), we note here that if 3 cos 2 = 0, then for our equation
θ
to hold, 3 sin 2 = 0 as well. Since no angles have both cosine and sine equal to zero,
θ
θ
θ
we are safe to divide both sides of the equation 3 sin 2 = 3 cos 2 by 3 cos 2 to get
θ
tan 2 = 1 which gives θ = π + 2πk for integers k . From these solutions, however, we
2
14 Again, we could have easily chosen to substitute these into r = 3 which would give −r = 3, or r = −3.
We obtain t...

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