Transcript Chapter 0

Chapter 6
Additional Topics
in Trigonometry
6.3 Polar Coordinates
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Objectives:
• Plot points in the polar coordinate system.
• Find multiple sets of polar coordinates for a given
point.
• Convert a point from polar to rectangular
coordinates.
• Convert a point from rectangular to polar
coordinates.
• Convert an equation from rectangular to polar
coordinates.
• Convert an equation from polar to rectangular
coordinates.
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Plotting Points in the Polar Coordinate System
The foundation of the polar coordinate system is a
horizontal ray that extends to the right. The ray is called
the polar axis. The endpoint of the ray is called the pole.
A point P in the polar coordinate system is represented by
an ordered pair of numbers (r , ). We refer to the ordered
pair P  (r , ) as the polar coordinates of P.
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Plotting Points in the Polar Coordinate System
(continued)
We refer to the ordered pair P  (r , ) as the polar
coordinates of P.
r is a directed distance from the pole to P.
 is an angle from the polar axis to the line segment from
the pole to P. This angle can be measured in degrees or
radians. Positive angles are measured counterclockwise
from the polar axis. Negative angles are measured
clockwise from the polar axis.
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Example: Plotting Points in a Polar Coordinate System
Plot the point with the following polar coordinates: (3,315)
Because 315° is a positive angle,
draw   315 counterclockwise
from the polar axis.
Because r = 3 is positive, plot the
point going out three units on the
terminal side of 
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(3,315)
315
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Example: Plotting Points in a Polar Coordinate System
Plot the point with the following polar coordinates: ( 2,  )
Because π is a positive angle,
draw    counterclockwise
from the polar axis.

 2,  
Because r = –2 is negative, plot the
point going out two units along
the ray opposite the terminal side
of 
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Example: Plotting Points in a Polar Coordinate System


Plot the point with the following polar coordinates:  1,  
2


Because  is a negative angle,
2
draw     clockwise
2
 1,   
from the polar axis.


Because r = –1 is negative, plot the
point going out one unit along the
ray opposite the terminal side of 
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

2

2
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Example: Finding Other Polar Coordinates for a Given
Point


Find another representation of  5,  in which
 4
r is positive and 2    4 .
We want r positive and 2    4 .
Add 2π to the angle and
do not change r.
 5,     5,   2 

 

 4  4

 5,  


4


 8   9 

  5,     5, 
 4 4   4 
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Example: Finding Other Polar Coordinates for a Given
Point


Find another representation of  5,  in which
 4
r is negative and 0    2 .
We want r negative and 0    2 .
Add π to the angle and
replace r with –r.
 5,     5,    

 

4
 4 

 5,  


4


 4   5 

  5, 
   5, 
4 4  
4 

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Example: Finding Other Polar Coordinates for a Given
Point


Find another representation of  5,  in which
 4
r is positive and 2    0.
We want r positive and 2    0.
Subtract 2π from the angle and
do not change r.
 5,     5,   2 

 

 4  4

 5,  


4


7 
 8  

  5,     5,  
4 
 4 4  
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Relations between Polar and Rectangular Coordinates
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Point Conversion from Polar to Rectangular Coordinates
To convert a point from polar coordinates ( r , ) to
rectangular coordinates (x, y), use the formulas
x  r cos and y  r sin  .
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Example: Polar-to-Rectangular Point Conversion
Find the rectangular coordinates of the point with the
following polar coordinates:
 3, 
(r , )   3, 
x  r cos  3cos   3(1)  3
y  r sin   3sin   3 0  0
The rectangular coordinates of
 3,  are (–3, 0).
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Example: Polar-to-Rectangular Point Conversion
Find the rectangular coordinates of the point with the
following polar coordinates: 

 10, 
6


(r , )   10, 
6

 3

x  r cos  10cos  10    5 3
6
 2 
  10  1   5
y  r sin   10sin
 
 2
6
 10,  
The rectangular coordinates of 
 are 5 3, 5 .
6


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
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Point Conversion from Rectangular to Polar Coordinates
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Example: Rectangular-to-Polar Point Conversion
Find polar coordinates of the point whose rectangular
coordinates are
1,  3 .


Step 1 Plot the point (x, y).
Step 2 Find r by computing the
distance from the origin to (x, y).
r  x2  y 2

y 3
 (1)   3
2
x 1

2
 1 3  4  2
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1,  3 
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Example: Rectangular-to-Polar Point Conversion
Find polar coordinates of the point whose rectangular
coordinates are
1,  3 .
y
Step 3 Find  using tan 
x
with the terminal side of


 passing through (x, y).
y  3
 3
tan   
1
x
x 1

5

3
y 3
1,  3 
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Example: Rectangular-to-Polar Point Conversion
Find polar coordinates of the point whose rectangular
coordinates are
1,  3 .




One representation of 1,  3 .
in polar coordinates is
5 

(r , )   2,  .
 3 
x 1

y 3
1,  3 
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Equation Conversion from Rectangular to Polar
Coordinates
A polar equation is an equation whose variables are r
and  To convert a rectangular equation in x and y to a
polar equation in r and  replace x with r cos and y
with r sin 
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Example: Converting Equations from Rectangular to
Polar Coordinates
Convert the following rectangular equation to a polar
equation that expresses r in terms of 
3x  y  6
The polar equation for
3r cos  r sin   6
is
3x  y  6
r (3cos  sin  )  6
6
r
3cos  sin 
6
r
3cos  sin 
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Example: Converting Equations from Rectangular to
Polar Coordinates
Convert the following rectangular equation to a polar
equation that expresses r in terms of 
r (r  2sin  )  0
x 2  ( y  1)2  1
2
2
r  0 r  2sin   0
 r cos    r sin   1  1
r  2sin 
r 2 cos 2   r 2 sin 2   2r sin   1  1
r 2 (cos2   sin 2  )  2r sin   1  1
r 2 (1)  2r sin   1  1
r 2  2r sin   1  1
r 2  2r sin   0
The polar equation for
x 2  ( y  1)2  1
is r  2sin 
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Equation Conversion from Polar to Rectangular
Coordinates
When we convert an equation from polar to rectangular
coordinates, our goal is to obtain an equation in which
the variables are x and y rather than r and  . We use
one or more of the following equations:
y
2
2
2
r cos  x r sin   y
r x y
tan  
x
To use these equations, it is sometimes necessary to
square both sides, to use an identity, to take the tangent
of both sides, or to multiply both sides by r.
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Example: Converting Equations from Polar to
Rectangular Form
Convert the polar equation to a rectangular equation in x
and y:
r4
2
r 2  x2  y 2
r4
r 2  16
r  4 or x2  y 2  16

0
x  y  16
2
2
3
2
The rectangular equation for r  4 is x2  y 2  16.
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Example: Converting Equations from Polar to
Rectangular Form
Convert the polar equation to a rectangular equation in x

3

3
2
and y:


4
4
4
3

4
3
tan   tan
4
tan   1
y
tan  
x

y
 1
x
y  x
3
or y   x
4

0
5
4
3
2
7
4
3
The rectangular equation for  
is y   x.
4
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Example: Converting Equations from Polar to
Rectangular Form
Convert the polar equation to a rectangular equation in x

and y: r  2sec
2
3

r  2sec
4
4
2
r
cos
r cos  2
x  2
r  2sec or x  2

0
5
4
3
2
7
4
The rectangular equation for r  2sec is x  2.
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Example: Converting Equations from Polar to
Rectangular Form
Convert the polar equation to a rectangular equation in x
and y: r  10sin 
r  10sin 
x 2  y 2  10 y  0
r 2  r (10sin  )
x 2  y 2  10 y  25  25
r 2  10r sin 
x2  ( y  5)2  25
r sin   y
The rectangular equation for r  10sin 
r 2  10 y
is x2  ( y  5)2  25.
2
2
2
r x y
2
2
x  y  10 y
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Example: Converting Equations from Polar to
Rectangular Form (continued)

2
Convert the polar equation
to a rectangular equation
in x and y: r  10sin 
The rectangular equation
for r  10sin 
is x2  ( y  5)2  25.
r  10sin  or x2  ( y  5)2  25
3
4

4
(0,5)

0
5
4
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7
4
3
2
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