**Unformatted text preview: **cos(2θ) ranges from 0 to 1 as well.7 From
this, we know r = ±4 cos(2θ) ranges continuously from 0 to ±4, respectively. Below we
graph both r = 4 cos(2θ) and r = −4 cos(2θ) on the θr plane and use them to sketch the
corresponding pieces of the curve r2 = 16 cos(2θ) in the xy -plane. As we have seen in earlier
7 Owing to the relationship between y = x and y =
former is deﬁned. √ x over [0, 1], we also know cos(2θ) ≥ cos(2θ) wherever the 808 Applications of Trigonometry
π
examples, the lines θ = π and θ = 34 , which are the zeros of the functions r = ±4
4
serve as guides for us to draw the curve as is passes through the origin. cos(2θ), y r θ= 4
1 π
2 π 3π
4 2 θ= 3 4 π
4 3π
4 π
4 1 x θ
2 3 4 −4 r = 4 cos(2θ) and
r = −4 cos(2θ )
As we plot points corresponding to values of θ outside of the interval [0, π ], we ﬁnd ourselves
retracing parts of the curve,8 so our ﬁnal answer is below.
y r θ= 4 π
4 π
2 3π
4 −4 π θ 3π
4 4 −4 θ= 4 π
4 x −4 r = ±4 cos(2θ)
in the θr-plane r2 = 16 cos(2θ)
in the xy...

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