AP CALCULUS BC
Section Number:
LECTURE NOTES
Topics: Polar Coordinates and Polar Graphs
MR. RECORD
Day: 40
10.4
Polar Coordinates
In a polar coordinate system, a fixed point O is called the pole or origin. The
polar axis is usually a horizontal ray directed toward the right from the pole.
The location of a point P in the polar coordinate system can be identified by
polar coordinates in the form  r ,  . If a ray is drawn from the pole through
P, the distance from the pole to point P is r . The measure of the angle
formed by OP and the polar axis is  . The angle can be measured in
degrees or radians.
It is important to consider both positive and negative values of r.
r  0 . Then  is the measure of any angle in
standard position that has OP as its terminal side.
Suppose
r  0 . Then  is the measure of any angle that
has the ray opposite OP as its terminal side.
Suppose
Example 1: Graph each point.
7 

a. P  1.5,

6 

Coordinate Conversions
Example 2: Converting Polar and Rectangular Coordinates.


b. Q  2,  
3

Example 2: Converting Rectangular and Polar Coordinates


a. Convert the polar coordinates  3, 
b. Convert the rectangular coordinates  1,1
6

to rectangular coordinates.
to polar coordinates.
Polar Graphs
Example 3: Graphing Polar Equations.
Graph each polar equation on the given polar coordinate planes.
a. r  2
b.  

3
Example 4: Graphing Polar Equations.
Sketch the graph of the polar equation r  3cos3 on the given polar coordinate plane.
c. r  sec
Special Polar Graphs
AP CALCULUS BC
Section Number:
LECTURE NOTES
Topics: Polar Coordinates and Polar Graphs
MR. RECORD
Day: 41
10.4
Slope and Tangent Lines
To find the slope of a tangent line to a polar graph, consider a differentiable function defined by r  f ( ) .
To find the slope in polar form, use the parametric equations
x  r cos  f   cos
and y  r sin   f   sin 
Using the parametric form of dy / dx given in the previous section, you have
dy dy / d

dx dx / d
THEOREM 10.11 SLOPE IN POLAR FORM
If f is a differentiable function of ϴ, then the slope of the tangent line to
f   cos   f    sin 

the graph of r  f   at the point  r ,  is
 f   sin   f    cos 
f   cos   f    sin 
dy dy / d


dx dx / d  f   sin   f    cos 
Example 5: Finding Horizontal and Vertical Tangent Lines.
a. Find the polar coordinates where the polar graph has horizontal and vertical tangents given
r  sin  , 0     .
b.
Find the polar coordinates where the polar graph has horizontal and vertical tangents given
r  2 1  cos 
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Part I
Part II
Theorem 10.11 has an important consequence. Suppose the graph of r  f   passes through the pole when
   and f     0 . Then the formula for dy / dx simplifies as follows:
f    sin   0 sin 
dy f    sin   f   cos 



 tan 
dx f    cos   f   sin  f    cos   0 cos 
So the line    is tangent to the graph at the pole  0,  
The graph f ( )  2 cos 3 illustrates this shortcut quite well.
THEOREM 10.12
TANGENT LINES AT THE POLE
If f    0 and f     0, then the line    is tangent at the pole to
the graph of r  f   .
Example 6: More Interesting Graphs of Polar Equations.
Graph each polar equation on the given polar coordinate planes.
6
a. r 
b. r  2cos 2 sec
2sin   3cos 
(a strophoid)