Transcript Math 112 Elementary Functions Section 4 Polar Coordinates and Graphs

Math 112
Elementary Functions
Chapter 7 – Applications of Trigonometry
Section 4
Polar Coordinates and Graphs
Rectangular (aka: Cartesian)
Coordinates
positive y-axis
(x, y)
negative x-axis
y
x
positive x-axis
origin
negative y-axis
For any
point there is
a unique
ordered pair
(x, y) that
specifies the
location of
that point.
Polar Coordinates
Is (r, )
unique for
every point?
(r, )
r
NO!

polar axis
pole
The angle  may be expressed in degrees or radians.
All of the
following refer
to the same
point:
(5, 120º)
(5, 480º)
(-5, 300º)
(-5, -60º)
etc ...
Polar Graph Paper
Locating and Graphing Points
90
120
60
(6, 75)
(5, 150)
(7, 0)
30
150
180
0
300)
(3,
(-3,-60)
120)
330
210
(-7, 180)
(-4, 30)
300
240
270
Converting Coordinates
Polar  Rectangular
Relationships between r, , x, & y
(r, )  (x, y)
r 2  x2  y2
y
r
x
Recommendation:
Find (r, ) where
r > 0 and
0 ≤  < 2 or 0 ≤  < 360.

y
tan  
x
RP
x  r cos
PR
y  r sin 
Examples: Converting Coordinates
Polar  Rectangular
x  r cos
(3, 210 )
 (3 cos 210 , 3 sin 210 )
  3
1 
  3 
, 3  
2
2 

 3 3 3

 
,
2 
 2
y  r sin 


  2, 
6




   2 cos ,  2 sin 
6
6


3
1
   2 
,  2  
2
2



  3, -1
Examples: Converting Coordinates
Polar  Rectangular
r x y
2
2
2
y
tan  
x
Quadrant I
(3, 7)
r  32  7 2  58
7
tan  
3
  tan 1
(3, 7)  ( 58 , 66.8 )
7
 66.8
3
Examples: Converting Coordinates
Polar  Rectangular
r x y
2
2
y
tan  
x
2
Quadrant II
(3, 7)
r  (3) 2  7 2  58
7
tan   
3
 7
  tan     66.8
 3
1
(3, 7)  ( 58 ,  66.8  180 )  ( 58 , 113.2 )
OR
(3, 7)  ( 58 ,  66.8 )

Examples: Converting Coordinates
Polar  Rectangular
r x y
2
2
2
y
tan  
x
Quadrant III
( 3,  7)
r  (3) 2  (7) 2  58
7
tan  
3
7
  tan
 66.8
3
1
(3,  7)  ( 58 , 66.8  180 )  ( 58 , 246.8 )
OR
(3,  7)  ( 58 , 66.8 )
Examples: Converting Coordinates
Polar  Rectangular
r x y
2
2
2
y
tan  
x
Quadrant IV
(3,  7)
r  32  (7) 2  58
7
tan   
3
 7
 3
  tan 1     66.8
(3,  7)  ( 58 ,  66.8  360 )  ( 58 , 293.2 )
OR
(3,  7)  ( 58 ,  66.8 )
Converting Equations
Polar  Rectangular
Use the same identities:
r x y
2
2
y
tan  
x
2
x  r cos
y  r sin 
Converting Equations
Polar  Rectangular
Replace all occurrences of x with r cos .
Replace all occurrences of y with r sin .
Simplify

Solve for r (if possible).
Converting Equations
Polar  Rectangular
Express the equation in terms of sine and cosine only.
If possible, manipulate the equation so that all occurrences of
cos  and sin  are multiplied by r.
Replace all occurrences of …
r cos 
with
x
r sin 
with
y
with
x2
r2
Or, if all else fails, use:
r  x2  y2
cos  
+
y2
Simplify (solve for y if possible)
sin  
x
x2  y2
y
x2  y2
Graphing Polar Equations
Reminder: How do you graph rectangular
equations?

Method 1:
Create a table of values.
 Plot ordered pairs.
 Connect the dots in order as x increases.


Method 2:
Recognize and graph various common forms.
 Examples: linear equations, quadratic equations, conics, …

The same basic approach can be applied to polar equations.
Graphing Polar Equations
Method 1: Plotting and Connecting Points
Create a table of values.
2. Plot ordered pairs.
3. Connect the dots in order as  increases.
1.
NOTE: Since most of these equations involve periodic
functions (esp. sine and cosine), at some point the graph
will start repeating itself (but not always).
Graphing Polar Equations
Method 2: Recognizing Common Forms
r=4
Circles

Centered at the origin: r = a


period = 360
Tangent to the x-axis at the origin: r = a sin 



radius: a
center: (a/2, 90)
a > 0  above
radius: a/2 period = 180
a < 0  below
Tangent to the y-axis at the origin: r = a cos 


center: (a/2, 90)
a > 0  right
r = 4 sin
radius: a/2 period = 180
a < 0  left
r = 4 cos
Graphing Polar Equations
Method 2: Recognizing Common Forms
Flowers (centered at the origin)

r = a cos n


radius: |a|
n is even  2n petals





petal every 180/n
period = 360
r = a sin n
r = 4 sin 2
n is odd  n petals


or
petal every 360/n
period = 180
cos  1st petal @ 0
sin  1st petal @ 90/n
r = 4 cos 3
Graphing Polar Equations
Method 2: Recognizing Common Forms
Spirals

Spiral of Archimedes: r = k

|k| large  loose
r=
|k| small  tight
r=¼
Graphing Polar Equations
Method 2: Recognizing Common Forms
Heart (actually: cardioid if a = b … otherwise: limaçon)

r = a ± b cos 
r = 3 + 3 cos 
or
r = 2 - 5 cos 
r = a ± b sin 
r = 3 + 2 sin 
r = 3 - 3 sin 
Graphing Polar Equations
Method 2: Recognizing Common Forms
Lines

Horizontal: y = k
 r sin  = k
 r = k csc 

Vertical:
 r cos  = h
 r = h sec 

Others:
x=h
ax + by = c 
r
c
a cos   b sin 
y = mx + b 
r
b
sin   m cos 
Graphing Polar Equations
Method 2: Recognizing Common Forms
Parabolas (w/ vertex on an axis)
a
r
1 cos 
r
3
1  cos 
r
7
1  cos 
a
r
1 sin 
r
5
1  sin 
NOTE: With these forms, the vertex will never be at the origin.
r
1
1  sin 