Transcript Document

Digital Lesson
9.6 Polar Coordinates
and Point Conversion
Day 1
9.6 Polar Coordinates
One way to give someone directions is to tell them to
go three blocks East and five blocks South.
Another way to give directions is to point and say “Go a
half mile in that direction.”
Polar graphing is like the second method of giving
directions. Each point is determined by a distance and
an angle.
r

Initial ray
A polar coordinate pair
 r , 
determines the location of
a point.
The polar coordinate system is formed by fixing a point, O,
which is the pole (or origin).
The polar axis is the ray constructed from O.
Each point P in the plane can be assigned polar coordinates (r, ).
P = (r, )
O
 = directed angle
Pole (Origin)
Polar
axis
r is the directed distance from O to P.
 is the directed angle (counterclockwise) from the polar axis
to OP.
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4
Plotting Points

2
b) Plot the point  3, 3 
4 


1
2
3
0
3,  34 
   3
4
3 units from
the pole
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3
2
5
Plotting Points
 
a) The point (r, )  2,  lies two units from the pole on the
3

2
terminal side of the angle    .
3
 
3
2, 
3
 

 
b) How else can we represent 2, 3 ?
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1
2
3
0
3
2
6
 
There are many ways to represent the point 2, 3 .
 2, 3    2,  53    2,  23    2, 43 
additional ways
to represent the

2
 2,  53 
 2, 3 

1
2
3
0
 
point 2, 
3
(r, )   r,  2n 
(r, )   r,  (2n  1) 
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3
2
7


 3,   ,
4

• Plot the polar point
then find two
additional polar representations of the point.

2

1
2
3
0
3
2
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8
The relationship between rectangular and polar
coordinates is as follows.
y
The point (x, y) lies on a
circle of radius r, therefore,
r2 = x2 + y2 .
(x, y)
(r, )
r
y
Pole

(Origin)
x
x
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Definitions of
trigonometric functions
cos   x
x  r cos
r
y
y  r sin 
sin  
r
y
tan  
x
9
x = r cos(θ)
Coordinate Conversion
y = r sin(θ)
cos   x
x  r cos
r
2 = x2 + y 2
y
r
y  r sin 
sin  
r
tan(θ) = y/x
y
2
2
2
(Pythagorean Identity)
tan  
r

x

y
x
Example:


Convert the point 4,   into rectangular coordinates.
3
x  r cos   4 cos    4 1  2
3
2


y  r sin   4 sin    4   3   2 3
3
 2 
 x, y   2, 2 3


  
 
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10
x = r cos(θ)
Example:
Convert the point
Example:
Convert the point
y = r sin(θ)
 2,   into rectangular coordinates.  2,0

3, 
6

into rectangular coordinates.
Example:
Convert the point
3 3
 ,

2 2 
  4.5,1.3 into rectangular coordinates.
 1.204,.  4.336
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11
Example:
Convert the point (1,1) into polar coordinates.
 x, y   1,1
y 1
tan     1
x 1
 
4
r  x 2  y 2  12  12  2
One set of polar coordinates is (r, ) 


Another set is (r, )   2, 5 .
4
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

2,  .
4
12
r2 = x2 + y2
tan(θ) = y/x
Example:
Convert the point (0,2) into polar coordinates.
 
 2, 
 2
Example:
Convert the point (-3,4) into polar coordinates.
5, 2.214
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13
r2 = x2 + y2
tan(θ) = y/x
Example:
Convert the point (-2,-3) into polar coordinates.

13, 4.124

Example:
Convert the point (-1,6) into polar coordinates.

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37, 1.735

14
Homework Day 1
9.6 pg. 680: 1-37odds
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15
Digital Lesson
9.6 Equation Conversion
Day 2
HWQ
Find 2 different sets of polar coordinates
for the rectangular point:
3, 1
Exact values only.

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
17
Equation Conversion
To convert a rectangular
equation to polar form, use:
To convert a polar equation
to rectangular form, use:
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x = r cos(θ)
y = r sin(θ)
r2 = x2 + y2
tan(θ) = y/x
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Equation Conversion
Example For the rectangular equation 3x + 2y = 4,
(a) convert to a polar equation,
(b) use a graphing calculator to graph the rectangular
equation, and
(c) use a graphing calculator to graph the polar
equation for 0°    360°.
Solution
(a) Let x = r cos  and y = r sin  to get
3r cos   2r sin   4 or
4
r
.
3cos   2sin 
Converting a Rectangular Equation to Polar Form
3
(b) Solve the rectangular equation for y to get y   x  2.
2
(c)
Converting a Polar Equation to Rectangular Form
Example For the polar equation r  4 ,
1  sin 
(a) convert to a rectangular equation,
(b) use a graphing calculator to graph the polar
equation for 0    2, and
(c) use a graphing calculator to graph the rectangular
equation.
Solution: Multiply both sides by the denominator.
2
2
2
2
2
2
x

y

4

y
r  r sin   4


x y y4

x 2  y 2  16  8 y  y 2
x  8  y  2
2

Converting a Polar Equation to Rectangular Form
(b) The figure shows
a graph with polar
coordinates.
(c) Solving x2 = –8(y – 2)
for y, we obtain
y  2  18 x 2 .
Equation Conversion
Ex: Convert the rectangular equation to polar
2
form: y  x
r  sec tan 
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24
Ex: Convert the rectangular equation to polar form:
x  y  16
2
2
r4
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25
Equation Conversion
Convert the polar equation r  4sin  into a rectangular
equation.
r  4sin 
Polar form
r 2  4r sin 
Multiply each side by r.
x2  y 2  4y
Substitute rectangular
coordinates.
x2  y 2  4y  0
x   y  2  4
2
2
Equation of a circle with
center (0, 2) and radius of 2
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27
Example:
Convert the polar equation r = 2 to
rectangular form.
x2  y 2  4
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28
Example:
Convert the polar equation to rectangular
form.   
3
tan   tan

3
y
 3
x
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y  3x
29
Example:
x = r cos(θ)
y = r sin(θ)
Convert the polar equation to rectangular
form. r  sec
r cos  sec cos
x  sec cos
x 1
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30
Example:
Convert from rectangular to polar form:
x y
2
3
r  cot  csc
2
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31
Example:
x = r cos(θ)
y = r sin(θ)
Convert the polar equation to rectangular
form. r  3cos 
2
3
9

2
x   y 
2
4

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32
x = r cos(θ)
Example:
y = r sin(θ)
Convert the polar equation to rectangular
form. r 2  sin 2
x
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2
y

2 2
 2 xy
33
Homework Day 2
9.6 pg. 680: 39-77 odds
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34
HWQ 3/24
• Convert the rectangular equation to polar
2
2
form: x  y  8 y  0
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35