chapter 8 – polar coordinates and parametric equations 8.2 - graphs of polar equations1
TRANSCRIPT
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18.2 - Graphs of Polar Equations
Section 8.2 Graphs of Polar
Equations
Chapter 8 – Polar Coordinates and Parametric Equations
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8.2 - Graphs of Polar Equations 2
Common Polar Curves: Circles and Spirals
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8.2 - Graphs of Polar Equations 3
Common Polar Curves: Limaçons
Orientation depends on the trigonometric function (sine or cosine) and the sign of b.
sin
cos
0, 0
r a b
r a b
a b
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8.2 - Graphs of Polar Equations 4
Common Polar Curves: Limaçons
Orientation depends on the trigonometric function (sine or cosine) and the sign of b.
sin
cos
0, 0
r a b
r a b
a b
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8.2 - Graphs of Polar Equations 5
Common Polar Curves: Roses
sin
cos
leaved of is odd
2 leaved of is even
r a n
r a n
n n
n n
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8.2 - Graphs of Polar Equations 6
Common Polar Curves: Roses
sin
cos
leaved of is odd
2 leaved of is even
r a n
r a n
n n
n n
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8.2 - Graphs of Polar Equations 7
Common Polar Curves: Lemniscates
Figure-eight-shaped curves
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8.2 - Graphs of Polar Equations 8
Example – pg. 553Match the polar equation with the
graphs labeled I – VI.
3. r = 3 cos
4. r = 3
5. r = 2 + 2 sin
6. r = 1 + 2 cos
7. r = sin 3
8. r = sin 4
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8.2 - Graphs of Polar Equations 9
SymmetryWhen working with polar equations, it is helpful to
use symmetry when graphing.
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8.2 - Graphs of Polar Equations 10
Tests for SymmetryIf a polar equation is
unchanged when we replace with – , then the graph is symmetric about the polar axis.
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8.2 - Graphs of Polar Equations 11
Tests for SymmetryIf a polar equation is
unchanged when we replace r with –r, then the graph is symmetric about the pole.
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8.2 - Graphs of Polar Equations 12
Tests for SymmetryIf a polar equation is
unchanged when we replace with – , then the graph is symmetric about the vertical line = /2.
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8.2 - Graphs of Polar Equations 13
Examples – pg. 553Test the polar equation for symmetry with respect to
the polar axis, the pole, and the line = /2.
10. 4 8cos
12. 5cos csc
r
r
2
514.
1 3cos
16. 9sin
r
r
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8.2 - Graphs of Polar Equations 14
Graphing Polar Equations
To sketch a polar curve whose graph, we plot points calculated for sufficiently many values of and join them in a continuous curve.
NOTE: This is how we first learned to graph functions in rectangular coordinates.
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8.2 - Graphs of Polar Equations 15
Example – pg. 553 # 23
Sketch a graph of the polar equation.r = -2 cos
r = -2 cos
0
/6
/4
/3
/2
(2)/3
(3)/4
(5)/6
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8.2 - Graphs of Polar Equations 16
Graphing Polar Equations
To sketch a polar curve whose graph, we can also sketch the graph in polar coordinates first. We can then use this graph to help us graph in the polar coordinate system.
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8.2 - Graphs of Polar Equations 17
Example – pg. 553 # 25
Sketch a graph of the polar equation.r = 2 – 2 cos
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8.2 - Graphs of Polar Equations 18
Examples – pg. 553Sketch a graph of the polar equation.
27. 3 1 sin
31. cos5
32. sin 4
r
r
r
2
36. 1 2cos
38. 4sin 2
39. , 0
r
r
r
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8.2 - Graphs of Polar Equations 19
Example – pg. 554Match the polar equation with
the graphs labeled I-IV. Give reasons for your answers.
49. r = cos( /2)
50. r = 1/ √( )
51. r = sin
52. r = 1 + 3 cos (3 )