Unit 6.4

1. 6.4
Polar
Coordinates
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Polar Coordinate System
 Coordinate Conversion
 Equation Conversion
 Finding Distance Using Polar Coordinates
… and why
Use of polar coordinates sometimes simplifies
complicated rectangular equations and they are useful in
calculus.
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3. The Polar Coordinate System
A polar coordinate system is a plane with a point O,
the pole, and a ray from O, the polar axis, as shown.
Each point P in the plane is assigned polar coordinates
as follows: r is the directed distance from O to P, and
 is the directed angle whose initial side is on the polar
axis and whose terminal side is on the line OP.
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4. Example Plotting Points in the Polar
Coordinate System
Plot the points with the given polar coordinates.
(a) P(2,  / 3) (b) Q(1, 3 / 4) (c) R(3,  45o)
Copyright © 2011 Pearson, Inc. Slide 6.4 - 4

5. Example Plotting Points in the Polar
Coordinate System
Plot the points with the given polar coordinates.
(a) P(2,  / 3) (b) Q(1, 3 / 4) (c) R(3,  45o)
Copyright © 2011 Pearson, Inc. Slide 6.4 - 5

6. Finding all Polar Coordinates of a Point
Let the point P have polar coordinates (r, ). Any
other polar coordinate of P must be of the form
(r,  2 n) or (r,  (2n 1) )
where n is any integer. In particular, the pole has
polar coordinates (0, ), where  is any angle.
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7. Coordinate Conversion Equations
Let the point P have polar coordinates (r, )
and rectangular coordinates (x, y). Then
x  r cos
y  r sin
r2  x2  y2
tan 
y
x
.
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8. Example Converting from Polar to
Rectangular Coordinates
Find the rectangular coordinates of the point with the
polar coordinates (2, 7 / 6).
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9. Example Converting from Polar to
Rectangular Coordinates
Find the rectangular coordinates of the point with the
polar coordinates (2, 7 / 6).
For point (2,7 / 6), r  2 and   7 / 6.
x  r cos y  r sin
x  2cos7 / 6 y  2sin7 / 6

x  2 
3
2
 

 

y  2 
1
2
 

 
x   3 y  1
The recangular coordinate is  3,1.
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10. Example Converting from
Rectangular to Polar Coordinates
Find two polar coordinate pairs for the point with
the rectangular coordinates (1, 1).
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11. Example Converting from
Rectangular to Polar Coordinates
Find two polar coordinate pairs for the point with
the rectangular coordinates (1, 1).
For the point (1, 1), x  1 and y  1.
r2  x2  y2 tan 
y
x
r2  12  (1)2 tan 
1
1
r2  2 tan  1
r   2   

4
 n
Two polar
coordinates
pairs are
2, 


4

 
  and
 2,

 .
3
4

 
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12. Example Converting from Polar
Form to Rectangular Form
Convert r  2csc to rectangular form and
identify the graph.
Support your answer with a polar graphing utility.
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13. Example Converting from Polar
Form to Rectangular Form
Convert r  2csc to rectangular form and
identify the graph.
Support your answer with a polar graphing utility.
r  2csc
r
 2
csc
r sin  2
y  2
The graph is the
horizontal line
y  2.
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14. Example Converting from Polar
Form to Rectangular Form
Convert x  22
 y  32
 13 to polar form.
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15. Example Converting from Polar
Form to Rectangular Form
Convert x  22
 y  32
x  22
 13 to polar form.
 y  32
 13
x2  4x  4  y2  6y  9  13
x2  y2  4x  6y  0
Substitute r2  x2  y2 ,
x  r cos , and y  r sin .
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16. Example Converting from Polar
Form to Rectangular Form
Convert x  22
 y  32
 13 to polar form.
Substitute r2  x2  y2 ,
x  r cos , and y  r sin .
r2  4r cos  6r sin  0
rr  4cos  6sin  0
r  0 or r  4cos  6sin
r  0 is a single point that is also on the other graph.
Thus the equation is r  4cos  6sin .
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17. Quick Review
1. Determine the quadrants containing the terminal
side of the angle:  4 / 3
2. Find a positive and negative angle coterminal
with the given angle:   / 3
3. Write a standard form equation for the circle with
center at (  6,0) and a radius of 4.
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18. Quick Review
Use the Law of Cosines to find the measure of the third
side of the given triangle.
4.
40º
8 10
5.
35º
6 11
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19. Quick Review Solutions
1. Determine the quadrants containing the terminal
side of the angle:  4 / 3 II
2. Find a positive and negative angle coterminal
with the given angle:   / 3 5 /3,  7 /3
3. Write a standard form equation for the circle
with center at (  6,0) and a radius of 4.
(x  6)2  y2  16
Copyright © 2011 Pearson, Inc. Slide 6.4 - 19

20. Quick Review Solutions
Use the Law of Cosines to find the measure of the
third side of the given triangle.
4.
40º
8 10
5.
35º
6 11
6.4
7
Copyright © 2011 Pearson, Inc. Slide 6.4 - 20