2.2. POLAR COORDINATES 65
Definition 2.3 Petal curves are curves with equations of the form:
r D cos.n/ or r D sin.n/;
where n is an integer.
If no restriction is placed on n, then the curve is traced out an infinite number of times. For odd
n, a domain of 2 Π0; / traces the curve once; when n is even, 2 Π0; 2/ is needed to trace
the entire curve once.
Example 2.27 Compare the curves r D sin.5/ and r D cos.5/ on the range 2 Π0; /.
(0,1)
(0,-1)
(1,0)(-1,0)
r D cos.5 /
(0,1)
(0,-1)
(1,0)(-1,0)
r D sin.5 /
˙
66 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS
e odd fact, that the minimal domain (to hit all the points) is twice as large when n is even,
is to some degree explained by the fact that, while petal curves with odd parameter n yield n
petals, when n is even we get 2n petals.
Example 2.28 Plot the polar function r D cos.4/.
Solution:
(0,1)
(0,-1)
(1,0)(-1,0)
See? We have n D 4 but 8 petals.
˙
It is possible to use values of the petal-determining parameter for polar curves that are not
integers, but then figuring out the minimal domain to plot the curve becomes problematic.
Example 2.29 Plot the polar function r D cos.1:5 / for 2 Π0; 4/.
Solution:
(0,1)
(0,-1)
(1,0)(-1,0)
˙
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