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θ
0
π
4
π
2
3π
4 π
5π
4
3π
2
7π
4 2π 2 r = 6 cos(θ)
6
√
32 (r, θ)
(6, 0)
√π
3 2, 4 0
√
−3 2 3 0, π
2
√ 3π
−3 2, 4 −6
√
−3 2 y (−6, π )
√ 5π
−3 2, 4 0
√
32
6 0, 3π
√ 72
π
3 2, 4 3 −3 (6, 2π ) For a review of these concepts and this process, see Sections 1.5 and 1.7. 6 x 11.5 Graphs of Polar Equations 799 Despite having nine ordered pairs, we only get four distinct points on the graph. For this reason,
we employ a slightly diﬀerent strategy. We graph one cycle of r = 6 cos(θ) on the θr plane3 and use
it to help graph the equation on the xy -plane. We see that as θ ranges from 0 to π , r ranges from 6
2
to 0. In the xy -plane, this means that the curve starts 6 units from the origin on the positive x-axis
(θ = 0) and gradually returns to the origin by the time the curve reaches the y -axis (θ = π ). The
2
arrows drawn in the ﬁgure below are meant to help you visualize this process. In the θr-plane, the
arrows are drawn from the θ-axis to the curve r = 6 cos(θ). In the xy -plane, each of these arrows
starts at the origin and is rotated through the corresponding angle θ, in accordance with how we
plot polar coordinates. It is a less-precise way to generate the graph than computing the actual
function values, but it is markedly faster.
y r
6 θ runs from 0 to π
2 3 π
2 π 3π
2 2π x θ −3 −6 Next, we repeat the process as θ ranges from π to π . Here, the r values are all negative. This
2
means that...

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