SalasSV_09_04 - 9.4 GRAPHING IN POLAR COORDINATES 539 25....

9.4 GRAPHING IN POLAR COORDINATES
We begin with the curve
r = θ , θ ≥ 0.
The graph is a nonending spiral, part of the famous spiral of Archimedes . The curve
is shown in detail from θ = 0 to θ = 2 π in Figure 9.4.1. At θ = 0, r = 0; at
θ =
1
4
π , r =
1
4
π ; at θ =
1
2
π , r =
1
2
π ; and so on.
The next examples involve trigonometric functions.
Example 1
Sketch the curve r = 1 − 2 cos θ .
SOLUTION
Since the cosine function is periodic with period 2 π , the curve r =
1 − 2 cos θ is a closed curve. We will draw it from θ = 0 to θ = 2 π . The curve
just repeats itself for values of θ outside the interval [0, 2 π ].

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CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS
x
O
r = θ θ 0
spiral of Archimedes
≥ ,
,
4
3
π
4
3
π
,
π π
,
4
5
π
4
5
π
,
2
3
π
2
3
π
Polar axis
[2 , 2 ]
π π
[ ]
,
4
1
π
4
1
π
,
2
1
π
2
1
π
,
4
7
π
4
7
π
Figure 9.4.1
We begin by compiling a table of values:
θ 0 π/ 4 π/ 3 π/ 2 2 π/ 3 3 π/ 4 π 5 π/ 4 4 π/ 3 3 π/ 2 5 π/ 3 7 π/ 4 2 π
r − 1 − 0. 41 0 1 2 2. 41 3 2. 41 2 1 0 − 0. 41 − 1
The values of θ for which r = 0 or | r | is a local maximum are as follows:
r = 0 at θ =
1
3
π ,
5
3
π for then cos θ =
1
2
; | r | is a local maximum at θ = 0, π , 2 π .
These five values of θ generate four intervals:
[0,
1
3
π ], [
1
3
π , π ], [ π ,
5
3
π ], [
5
3
π , 2 π ].
We sketch the curve in four stages. These stages are shown in Figure 9.4.2.
[ – 1, 0]
polar axis
0 ≤ ≤ θ
1
3
π
[ – 1, 0]
0 ≤ ≤ θ π
[3, ] π
[ – 1, 0]
polar axis
0 ≤ ≤ θ
5
3
π
0 ≤ ≤ 2 θ π
polar axis
polar axis
Figure 9.4.2
As θ increases from 0 to
1
3
π , cos θ decreases from 1 to
1
2
and r = 1 − 2 cos θ increases
from − 1 to 0.
As θ increases from
1
3
π to π , cos θ decreases from
1
2
to − 1 and r increases from 0 to 3.
As θ increases from π to
5
3
π , cos θ increases from − 1 to
1
2
and r decreases from 3 to 0.
Finally, as θ increases from
5
3
π to 2 π , cos θ increases from
1
2
to 1 and r decreases
from 0 to − 1.
As we could have read from the equation, the curve is symmetric about the x -axis
[ r ( − θ ) = 1 − 2 cos ( − θ ) = 1 − 2 cos θ = r ( θ )].
Example 2
Sketch the curve r = cos 2 θ , 0 ≤ θ ≤ 2 π .

9.4 GRAPHING IN POLAR COORDINATES
541
SOLUTION
As an alternative to compiling a table of values as we did in Example 1, we
refer to the graph of cos 2 θ in rectangular coordinates for the values of r . See Figure
9.4.3. The values of θ for which r is zero or has an extreme value are as follows:
r = 0 at θ =
π
4
,
3 π
4
,
5 π
4
,
7 π
4
; local maxima/minima at θ = 0,
π
2
, π ,
3 π
2
, 2 π .
θ cos 2
1
– 1
π
θ π 2
4
π
2
π
4
π 3
4
π 5
4
π 7
2
π 3
Figure 9.4.3
In Figure 9.4.4 we sketch the curve in eight stages.
0 ≤ ≤ θ
1
4
π
θ
1
4
π =
0 ≤ ≤ θ
1
2
π 0 ≤ ≤ θ π
0 ≤ ≤ θ
3
2
π 0 ≤ ≤ θ π 2 0 ≤ ≤ θ
5
4
π
0 ≤ ≤ θ
3
4
π
0 ≤ ≤ θ
7
4
π
Figure 9.4.4
Example 3 Figure 9.4.5 shows four cardioids , heart-shaped curves. Rotation of
r = 1 + cos θ by
1
2
π radians, measured in the counterclockwise direction, gives
r = 1 + cos ( θ −
1
2
π ) = 1 + sin θ .
r = 1 + cos θ r = 1 + sin θ r = 1 – cos θ r = 1 – sin θ
Figure 9.4.5
Rotation by another
1
2
π radians gives
r = 1 + cos ( θ − π ) = 1 − cos θ .

542
CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS
Rotation by yet another
1
2
π radians gives
r = 1 + cos ( θ −
3
2
π ) = 1 − sin θ .
Notice how easy it is to rotate axes in polar coordinates: each change
cos sin cos sin θ θ θ θ → → − → −
represents a counterclockwise rotation by
1
2
π radians.