Intro to Polar Coordinates
Lesson 6.5A
Points on a Plane

Rectangular coordinate system




Represent a point by two distances from the
(x, y)
origin
•
Horizontal dist, Vertical dist
Also possible to represent different ways
Consider using dist from origin, angle
formed with positive x-axis (r, θ)
•
θ
r
2
Plot Given Polar Coordinates

Locate the following
 
A   2, 
 4
 2 
B   4,

 3 
3 

C   3, 
2 

5 

D   1, 

4 

3
Find Polar Coordinates

What are the coordinates for the given
points?
• B
• A
•D
•C
•A=
•B=
•C=
•D=
4
Converting Polar to Rectangular

Given polar coordinates (r, θ)

Change to rectangular
r

By trigonometry



4
y
θ
x = r cos θ
y = r sin θ
Try A   2,  
•
x
=
( ___, ___ )
5
Converting Rectangular to Polar

Given a point (x, y)


r
Convert to (r, θ)
  tan

By trigonometry

Try this one … for (2, 1)

y
θ
By Pythagorean theorem
r 2 = x2 + y2

•
x
1
y
x
r = ______
θ = ______
6
Polar Equations


States a relationship between all the
points (r, θ) that satisfy the equation
Example
r = 4 sin θ

Resulting values
Note: for (r, θ)
θ in
degrees
It is θ (the 2nd element
that is the independent
variable
7
Graphing Polar Equations

Set Mode on TI calculator


Mode, then Graph => Polar
Note difference of Y= screen
8
Graphing Polar Equations


Also best to keep
angles in radians
Enter function in
Y= screen
9
Graphing Polar Equations

Set Zoom to
Standard,

then Square
10
Try These!

For r = A cos B θ




Try to determine what affect A and B have
r = 3 sin 2θ
r = 4 cos 3θ
r = 2 + 5 sin 4θ
Experiment with
Polar Function
Spreadsheet
11
Assignment A



Lesson 6.5A
Page 424
Exercises 1 – 47 odd
12
Polar Coordinates
Lesson 6.5B
Write Polar Equation in Rectangular
Form

Given r = 2 sin θ


Use definitions


Write as rectangular
equation
And identities
(see inside back cover)
x  r cos 
y  r sin 
y
 tan 1 
x
r 2  x2  y 2
Graph the given equation for clues
14
Write Polar Equation in Rectangular
Form

Given r = 2 sin θ

We know

Thus

And
r
y
 sin  
2
r
2
r  2y
x2  y 2  2 y
15
Write Rectangular Equation in Polar
Form

Consider 2x – 3y = 6
x  r cos 
y  r sin 

As before, use
definitions
2  r cos   3  r sin   6
y
 tan 1 
x
r 2  x2  y 2
r  2 cos   3sin    6
6
r
 2 cos   3sin  
16
Assignment B



Lesson 6.5B
Page 424
Exercises 49 – 73 odd
17