Transcript 10.4 Polar

10
Conics, Parametric Equations,
and Polar Coordinates
Copyright © Cengage Learning. All rights reserved.
10.4
Polar Coordinates and
Polar Graphs
2015
Copyright © Cengage Learning. All rights reserved.
Objectives
 Understand the polar coordinate system.
 Rewrite rectangular coordinates and equations in
polar form and vice versa.
 Sketch the graph of an equation given in polar form.
 Find the slope of a tangent line to a polar graph.
 Identify several types of special polar graphs.
10.4
Polar Coordinates
Greg Kelly, Hanford High School, Richland, Washington
Polar Coordinates
Polar Coordinates
You may represent graphs as collections of points (x, y) on the
rectangular coordinate system.
The corresponding equations for these graphs have been in either
rectangular or parametric form.
In this section you will study a coordinate system called the polar
coordinate system.
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.

Polar Coordinates
To form the polar coordinate system in the plane,
fix a point O, called the pole (or origin), and construct
from O an initial ray called the polar axis, as shown
in Figure 10.36.
Figure 10.36
Then each point P in the plane can be assigned
polar coordinates (r, θ), as follows.
r = directed distance from O to P
θ = directed angle, counterclockwise from polar axis
to segment OP
Figure 10.37 shows three points on the polar coordinate
system.
Figure 10.37
Some curves are easier to describe with polar coordinates:
r a
(Circle centered at the origin)
  o
(Line through the origin)
1 r  2
0  

2

Coordinate Conversion
To establish the relationship between polar and rectangular
coordinates, let the polar axis coincide with the positive
x-axis and the pole with the origin, as shown in Figure 10.38.
Because (x, y) lies on a circle of
radius r, it follows that r2 = x2 + y2.
Moreover, for r > 0 the definitions
of the trigonometric functions
imply that
Figure 10.38
and
If r < 0, you can show that the same relationships hold.
Coordinate Conversion
Polar Coordinates are not unique.
More than one coordinate pair can refer to the same point.
 2,30 
o
2
210 o
30
o
150o
o
  2, 210

  2, 150
o

All of the polar coordinates of this point are:
o
o
2,30

n

360


o
o

2,

150

n

360


n  0,  1,  2 ...

Polar-to-Rectangular Conversion
a. Convert the point (2, π) to rectangular form.
The rectangular coordinates are (x, y) = (–2, 0).

b. Convert the point  3,  to rectangular form.
6

The rectangular coordinates are (x, y) =
Rectangular to-Polar Conversion
a. Convert the point (1,1) to polar form.
One set of polar coordinates is (r, ) 


Another set is (r, )   2, 5 .
4

b. Convert the point (-3,4) to polar form.
5, 2.214

2,  .
4
Equation Conversion
a. Convert the equation
to polar form.
tan 
r
cos 
b. Convert the equation
to rectangular form.
Polar Graphs
Tests for Symmetry:
x-axis: If (r, ) is on the graph, so is (r, -).
1
r
r  2cos

0

1
2
r
-1

Tests for Symmetry:
y-axis: If (r, ) is on the graph, so is (r, -) or (-r, -).
2
r
r  2sin
1
 
-1
r

0
1


Tests for Symmetry:
origin: If (r, ) is on the graph, so is (-r, ) or (r, +) .
2
tan 
r
cos 
1
r

-2
0
-1
 
r
1
2
-1
-2

Tests for Symmetry:
If a graph has two symmetries, then it has all three:
2
1
r  2cos  2 
-2
-1
0
1
2
-1
-2

Example 3 – Graphing Polar Equations
Describe the graph of each polar equation. Confirm each
description by converting to a rectangular equation.
Example 3(a) – Solution
The graph of the polar equation r = 2 consists of all points
that are two units from the pole.
In other words, this graph
is a circle centered at the origin
with a radius of 2.
[See Figure 10.41(a).]
You can confirm this by using the
relationship r2 = x2 + y2
to obtain the rectangular equation
Figure 10.41(a)
Example 3(a) – Solution
The graph of the polar equation θ = π/3 consists of all
points on the line that makes an angle of π/3 with the
positive x-axis. [See Figure 10.41(b).]
You can confirm this by using the
relationship tan θ = y/x to obtain the
rectangular equation
Figure 10.41(b)
Example 3(c) – Solution
The graph of the polar equation r = sec θ is not evident by
simple inspection, so you can begin by converting to
rectangular form using the relationship r cos θ = x.
From the rectangular equation,
you can see that the graph is a
vertical line.
Figure 10.41(c)
Polar Graphs
The graph of
shown in Figure 10.42 was produced with a
graphing calculator in parametric mode. This equation was
graphed using the parametric equations
with the values of θ varying
from –4π to 4π.
This curve is of the form r = aθ
and is called a spiral of Archimedes.
Figure 10.42
Slope and Tangent Lines
Not tested, optional
To find the slope of a polar curve:
1. Write the equation in parametric form using:
x = r cos θ and y = r sin θ
2. Differentiate x and y with respect to θ.
dy
dy
 d
dx
dx
d
d
r sin 
 d
d
r cos 
d
r  sin   r cos 

r  cos   r sin 
We use the product rule here.
To find the slope of a polar curve:
Product rule
dy
dy
 d
dx
dx
d
d
r sin 
 d
d
r cos 
d
r  sin   r cos 

r  cos   r sin 
dy r  sin   r cos 

dx r  cos   r sin 

Example:
r  sin
r  1  cos
dy r  sin   r cos 

dx r  cos   r sin 
sin  sin   1  cos   cos 
Slope 
sin  cos  1  cos   sin 
sin   cos  cos 

sin  cos  sin   sin  cos
2
2
sin   cos   cos

2sin  cos  sin 
2
2
 cos 2  cos

sin 2  sin 

Slope and Tangent Lines
Figure 10.45
Slope and Tangent Lines
From Theorem 10.11, you can make the following
observations.
1. Solutions to
yield horizontal tangents, provided
that
2. Solutions to
yield vertical tangents, provided
that
If dy/dθ and dx/dθ are simultaneously 0, no conclusion can
be drawn about tangent lines.
Example 5 – Finding Horizontal and Vertical Tangent Lines
Find the horizontal and vertical tangent lines of
r = sin θ, 0 ≤ θ ≤ π.
Solution:
Begin by writing the equation in parametric form.
x = r cos θ = sin θ cos θ
and
y = r sin θ = sin θ sin θ = sin2 θ
Next, differentiate x and y with respect to θ and set each
derivative equal to 0.
Example 5 – Solution to finding horizontal and vertical
tangent lines to r = sin θ.
So, the graph has vertical tangent
lines at
and
and it has horizontal tangent lines
at (0, 0) and (1, π/2).
Figure 10.46
Ex: Find any lines tangent to the pole for f    2cos3
Solution:
 
5
Since f    2 cos 3  0 when   , , and
,
6 2
6
and the derivative f     6sin 3  0 for any of these values,
 
5
f   has three tangent lines at   , , and .
6 2
6
Special Polar Graphs
Special Polar Graphs
Several important types of graphs have equations that are
simpler in polar form than in rectangular form. For example,
the polar equation of a circle having a radius of a and
centered at the origin is simply r = a. Several other
types of graphs that have simpler equations in polar form
are shown below.
Special Polar Graphs
Homework
• Section 10.4 Day 1: pg. 736, #1-51 odd
• Day 2: MMM pgs 162-163