Honors Pre-Calculus
Chapter 8
Polar Coordinates
Mrs. Kendall
Chapter 8 – Assignment Guide
3/18
M:
QTR EXAM
T:
p. 590 9-37 eoo, 39, 45-51 o
W:
p. 590 57-71 o, p. 607 1-15 o, 31-59 eoo (calc & sketch)
R:
Review
F:
8.1 - 8.2 QUIZ
SPRING BREAK!!! :)
Honors Pre-Calculus
8.1
Introduction to Polar Coordinates
Learning Targets: Students will be able to graph polar coordinates and rewrite polar coordinates to
rectangular and vice- versa.
.
Until now, we would plot points in a plane using the
RECTANGULAR COORDINATE SYSTEM.
This is where a set of ordered pairs (x, y) are found
on the x-axis and y-axis which intersect at the origin
(0, 0).
Plot:
A (2, -3)
B (-4, -1)
There are other types of coordinate systems out
there. We are going to study:
POLAR COORDINATE SYSTEM
The polar coordinate system pairs  r ,  whose coordinates are found at a distance r (radius) along the polar
axis from fixed point (pole) at an angle of rotation .
For each ordered pair  r ,  , if:
rotated axis

Pole
+
-
+r
-r
rotate counterclockwise
rotate clockwise
plot point along rotated axis
plot point opposite of rotated axis
Polar Axis
For simplicity, relations between polar and rectangular:
Pole = Origin
Polar axis = positive x-axis
Plot each point given in
polar coordinates.
10.
A (4, 270°)
14.
B  5,
 5 

 3 
16.
C (-3, 120°)
18.
D  3, 


3 

4 
Find 3 other ordered pairs that
have the same location as the
given point. (You may not rotate
360 degrees/2π.)
22.
 3 
 4,

 4 
Converting between polar and rectangular:
Find all of the equations that relate polar and
rectangular coordinates.
P
Find the rectangular coordinates of each point.
34.
(5, 300°)
38.
3 

 3,  
4 

52.
 2, 2 3 
Find the polar coordinates of each point.
50.
(-3, 3)
Honors Pre-Calculus
8.1 & 8.2
Polar Equations and Graphs
Learning Targets: Students will be able to convert polar equations to rectangular equations and vice-versa.
Students will also be able to graph different types of polar graphs giving a minimum of four points.
r
x  r cos
cos  
x
r
y  r sin 
sin  
y
r
x2  y 2  r 2  r   x2  y 2
tan  
y
x
y

x
8.1 – continued
Write each equation using POLAR coordinates.
(for consistency, always solve for r  or r 2  in terms of )
58.
x2  y 2  x
60.
y2  2x
72.
r
Write each equation using RECTANGULAR coordinates.
(for consistency, always set final equation = 0, in terms of x and y).
70.
r4
3
3  cos 
8.2 – Polar Equations and Graphs
Circles
r#
center @ pole
tangent to pole,
center @ (0, a)
tangent to pole
center @ (a, 0)
r  2a sin 
r  2a cos
Lines
through the pole
horizontal:
vertical:
Other
 #
r sin   #
r cos  #
Cardioid:
ex:
r  2  2cos
Limaçon:
…without an inner loop
ex:
r  3  2cos
…with an inner loop
ex:
r  1  2cos
Rose:
ex:
r  3cos 2
Lemniscate:
When graphing these, you must put at least four point (at the quadrantal
values).
Example:
ex:
r 2  4sin 2
r  2  2cos
Spiral:
ex:

r e
5
Honors Pre-Calculus
Review 8.1 & 8.2
Polar Coordinates
Learning Targets: Students will be able to graph polar coordinates and rewrite polar coordinates to
rectangular and vice- versa.
.
Plot and label the point on the graph provided.
Find 3 other ordered pairs that have the same
location as the given point.
4 
7 
2 


 7 

1.
A  3, 
2.
B  2, 
5.
6.


 2,

 3,

3 
6 
3 


 4 

3 

 
3.
C  1, 
4.
D  5, 
4 

 2
Convert the given polar coordinates into
rectangular coordinates.
7.
Convert the given coordinates into polar
coordinates.
9.
 5, 5
10.
 0, 3
.
5 

 5, 

6 

8.
5 

 3,

3 

Learning Targets: Students will be able to convert polar equations to rectangular equations and vice-versa.
Students will also be able to graph different types of polar graphs giving a minimum of four points.
Convert the given rectangular equation into a
polar equation.
x 2  y 2  4 x
11.
3x  5 y  2
12.
Convert the given polar equation into a
rectangular equation.
13.
r 7
16.

14.
Graph. (Be sure to plot all of the key points.)
r  3  3sin 
15.


 


 
5.  2, 
Answers
1-4
6.  3, 
D
4  5 
 3, 
 3,

3 
3  3 
 5 3 5
,  
2
2

C
B


3
3 
5 
 2,  2, 

4 
4 
4 
7.  

A

9.  5 2,
5 

4 
11. r  4cos
13. x 2  y 2  49  0
 3 3 3
 2 2 


8.   ,
 3 

 2 
10.  3,
2
3cos   5sin 
2
14. x  y 2  8 x  0
12. r 
15-16.
r  8cos
Graph Paper