James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

662 Chapter 10 Parametric Equations and Polar Coordinates
r
2
1
0
π
2
π
3π
2
2π ¨
FIGURE 10
r − 1 1 sin in Cartesian coordinates,
0 < < 2
Example 7 Sketch the curve r − 1 1 sin .
SOLUtion Instead of plotting points as in Example 6, we first sketch the graph of
r − 1 1 sin in Cartesian coordinates in Figure 10 by shifting the sine curve up one
unit. This enables us to read at a glance the values of r that correspond to increasing
values of . For instance, we see that as increases from 0 to y2, r (the distance from
O) increases from 1 to 2, so we sketch the corresponding part of the polar curve in
Figure 11(a). As increases from y2 to , Figure 10 shows that r decreases from 2 to
1, so we sketch the next part of the curve as in Figure 11(b). As increases from to
3y2, r decreases from 1 to 0 as shown in part (c). Finally, as increases from 3y2
to 2, r increases from 0 to 1 as shown in part (d). If we let increase beyond 2 or
decrease beyond 0, we would simply re trace our path. Putting together the parts of the
curve from Figure 11(a)–(d), we sketch the complete curve in part (e). It is called a
cardioid because it’s shaped like a heart.
¨= π 2
¨= π 2
2
O 1 ¨=0
¨=π
O
¨=π
O
O
¨=2π
O
¨= 3π 2
¨= 3π 2
(a) (b) (c) (d) (e)
FIGURE 11 Stages in sketching the cardioid r − 1 1 sin
n
tec Module 10.3 helps you see how
polar curves are traced out by showing
animations similar to Figures 10–13.
Example 8 Sketch the curve r − cos 2.
SOLUTION As in Example 7, we first sketch r − cos 2, 0 < < 2, in Cartesian
coordinates in Figure 12. As increases from 0 to y4, Figure 12 shows that r
decreases from 1 to 0 and so we draw the corresponding portion of the polar curve in
Figure 13 (indicated by ). As increases from y4 to y2, r goes from 0 to 21. This
means that the distance from O increases from 0 to 1, but instead of being in the first
quadrant this portion of the polar curve (indicated by ) lies on the opposite side of the
pole in the third quadrant. The remainder of the curve is drawn in a similar fashion,
with the arrows and numbers indicating the order in which the portions are traced out.
The resulting curve has four loops and is called a four-leaved rose.
r
¨= π 2
1
!
$
% *
¨= 3π 4
$
&
^
!
¨= π 4
π
4
π
2
3π
4
π
@ # ^ &
5π
4
3π
2
7π
4
2π
¨
¨=π
%
@ #
*
¨=0
FIGURE 12
r − cos 2 in Cartesian coordinates
FIGURE 13
Four-leaved rose r − cos 2
n
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