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Graphs of Polar Equations

Dr .Hayk MelikyanDepartmen of Mathematics and CS

melikyan@nccu.edu

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Recall that a polar equation is an equation whose variables are r and è. The graph of a polar equation is the set of all points whose polar coordinates satisfy the equation. We use polar grids like the one shown to graph polar equations. The grid consists of circles with centers at the pole. This polar grid shows five such circles. A polar grid also shows lines passing through the pole, In this grid, each fine represents an angle for which we know the exact values of the trigonometric functions.

0

6˝4

32

32

43

65

67

45

34

35 4

7 6

11

23

42

Using Polar Grids to Graph Polar Equations

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˝ 0

64

˝32

32

43

65

67

45

34

35 4

7 6

11

23

42

Solution We construct a partial table of coordinates using multiples of 6. Then we plot the points and join them with a smooth curve, as shown.

(-4, )4 cos ˝ = 4(-1) = -4˝

(-3.5,5 /6)4 cos5 /6 = 4(- 3/2)=-2 3=-3.55/6

(-2, 2 /3)4 cos2 /3 = 4(- 1/2) = -22 /3

(0, /2)4 cos /2 = 4 • 0 = 0 /2

(2, /3)4 cos /3 = 4 • 1/2 = 4 /3

(3.5, /6)4 cos /6 = 4 •3/2=2 3=3.5 /6

(4, 0)4 cos 0 = 4 • 1 = 40

(r, )r 4 cos

(0, /2)

(2, /3)

(3.5, /6)

(4, 0) or (-4, )

(-3.5, 5 /6)

(-2, 2 /3)

Text Example

Graph the polar equation r 4 cos with in radians.

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The graphs ofr = a cos and r = a sin

Are circles.r = a cos r = a sin

The graphs ofr = a cos and r = a sin

Are circles.r = a cos r = a sin

˝ 0

/2

3 /2

˝ 0

/2

3 /2

a

a

Circles in Polar Coordinates

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Solution We apply each of the tests for symmetry.

Polar Axis: Replace by in r 1 cos :r 1 cos () Replace by in r 1 cos .

r 1 cos The cosine function is even: cos ( ) cos .

Because the polar equation does not change when is replaced by , the graph is symmetric with respect to the polar axis.

Text Example

Check for symmetry and then graph the polar equation: r 1 cos .

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The Line 2: Replace (r, ) by (r, ) in r 1 cos :r 1 cos() Replace r by r and by – in r 1 cos( ).

r 1 – cos cos( ) cos .

r cos 1 Multiply both sides by 1.

Because the polar equation r 1 cos changes to r cos 1 when (r, ) is replaced by (r, ), the equation fails this symmetry test. The graph may

of may not be symmetric with respect to the line 2.

Solution

The Pole: Replace r by r in r 1 cos :r 1 – cos Replace r by –r.

r cos 1 Multiply both sides by 1.

Because the polar equation r 1 cos changes to r cos 1 when r is replaced by r, the equation fails this symmetry test. The graph may or may not be symmetric with respect to the pole.

Text Example cont.

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Now we are ready to graph r 1 cos . Because the period of the cosine function is 2r, we need not consider values of beyond 2. Recall that we discovered the graph of the equation r 1 cos has symmetry with respect to the polar axis. Because the graph has symmetry, we may be able to obtain a complete graph without plotting points generated by values of from 0 to 2. Let's start by finding the values of r for values of from 0 to .

Solution

21.871.501.000.500.130r

5/62/32/3/60

The values for r and are in the table. Examine the graph. Keep in mind that the graph must be symmetric with respect to the polar axis.

0

64

3˝2

32

43

65

67

45

34

35 4

7 6

11

23

21

Text Example cont.

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Thus, if we reflect the graph from the last slide about the polar axis, we will obtain a complete graph of r 1 cos , shown below.

Solution

0

64

32

32

43

65

67

45

34

35 4

7 6

11

23

1 2

Text Example cont.

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The graphs ofr a b sin , r a b sin , r a b cos , r a b cos , a > 0, b > 0

are called limacons. The ratio ab determines a limacon's shape.

Inner loop if ab < 1 Heart shaped if ab 1 Dimpled with no inner No dimple and no inner

and called cardiods loop if 1< ab < 2 loop if ab 2.

The graphs ofr a b sin , r a b sin , r a b cos , r a b cos , a > 0, b > 0

are called limacons. The ratio ab determines a limacon's shape.

Inner loop if ab < 1 Heart shaped if ab 1 Dimpled with no inner No dimple and no inner

and called cardiods loop if 1< ab < 2 loop if ab 2.

0

2

23

0

2

23

0

2

23

0

2

23

Limacons

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Example

Graph the polar equation y= 2+3cos

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Example

Graph the polar equation y= 2+3cos

Solution:

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The graphs ofr a sin n and r a cos n, a does not equal 0,

are called rose curves. If n is even, the rose has 2n petals. If n is odd, the rose has n petals.

r a sin 2 r a cos 3 r a cos 4 r a sin 5 Rose curve Rose curve Rose curve Rose curve

with 4 petals with 3 petals with 8 petalswith 5 petals

The graphs ofr a sin n and r a cos n, a does not equal 0,

are called rose curves. If n is even, the rose has 2n petals. If n is odd, the rose has n petals.

r a sin 2 r a cos 3 r a cos 4 r a sin 5 Rose curve Rose curve Rose curve Rose curve

with 4 petals with 3 petals with 8 petalswith 5 petals

n = 2

a

0

2

23

0

2

23

n = 5

a

0

2

23

n = 4

a

0

2

23

n = 3

a a

Rose Curves

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Example

Graph the polar equation y=3sin2

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Example

Graph the polar equation y=3sin2

Solution:

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Lemniscate:

r2 = a2 cos 2

Lemniscates

The graphs of r2 = a2 sin 2 and r2 = a2 cos 2 are called lemniscates