Mathematical Investigations IV
Name:
Mathematical Investigations IV
Polar Coordinates-Out and Around
An Introduction
By now, you are adept at graphing functions in Cartesian (x, y) coordinates. Given a set of axes
and an ordered pair (p, q), you know how to plot the corresponding point: move horizontally p
units and vertically q units – make a dot and label it. Although many situations map well onto a
rectangular grid (for example, the streets of Chicagoland), there are some situations that have
circular (or spherical) symmetry instead. Think of planets orbiting the sun, of satellites orbiting
the earth, of circles, spheres, and right circular cylinders in everyday life. Polar coordinates give
us a more elegant way of expressing position in these cases.
Polar graph paper looks different from rectangular graph paper. Instead of having evenly spaced
vertical and horizontal lines, polar graph paper has a pole (point at the center), concentric circles
and radial lines containing the pole.
Cartesian
(rectangular)
axes
Polar axes
(r, )
(x, y)
In Cartesian coordinates, we locate a point, (x, y), by moving x-distance along the x-axis and then
moving y-distance parallel to the y-axis. In polar coordinates (r, ), a point is located by its
distance from the pole and the angle it determines with the positive x-axis. For example, verify
the position of each point below.
P(2, /6)
T(–3, /2)
 S
V 
Q(1, 0)
U(–2, 2)
U
R(1.5, –3/4)
V(3, –5/4)

1
 P
Q
2
3
 R

S(3, /2)
T
What do you know about the coordinates of the "pole"?
Polar 1.1
Rev. F05
Mathematical Investigations IV
Name:
1.
Use the graph at the right.
a.
2.
Plot the following points:
A(2, /6)
A (–2, /6)
B(1, –5/6)
B  (–1, –5/6)
C(3, 5/3)
C  (–3, 2/3)
D(1.5, /4)
D  (–1.5, –/4)
b.
What do you notice about points C and C  ?
Explain.
c.
Can a point in Cartesian coordinates be represented by two different ordered pairs
in polar coordinates? If "Yes," give an example. If "No," explain why.
d.
Explain how you plot a point where r is negative.
Find alternative coordinate pairs for the following. Use positive/negative values for r and
 as indicated.
(r+, –)
(r–, +)
(r–, –)
A( 2, 5/6)
B( 3, 4/3)
3.
or
or
(
,
)
or (
,
)
or
(
,
)
(
,
)
or (
,
)
or
(
,
)
Generalize: If A(2, 5/6) and P(r, ) are the same point, what relations exist between the
"2" and the "r", and between the "5/6" and the "", if
a.
r > 0?
b.
r < 0?
Polar 1.2
Rev. F05
Mathematical Investigations IV
Name:
4.
Considering the previous results, what can you say about polar coordinates as opposed to
rectangular coordinates?
5.
We need to examine the precise relationship between the ordered pairs (x, y) and (r, ),
which are usually distinguished by context.


Complete, according to the picture:
(x, y)
cos() =
so x =
sin() =
so y =
(r,  )
r
y

Also, in terms of x and y,
x
r2 =
and
tan() =



6.
Convert the following from polar to rectangular, using a calculator if necessary.
Note: Plotting the point may be helpful. Exact where possible; otherwise, round to tenths.
(4, /3)
(–6, 3/4)
(3, 108°)
(–10, –/5)
(–12, 4) (radians)
(3, 220°)
Polar 1.3
Rev. F05
Mathematical Investigations IV
Name:
7.
Using radian measure, convert the following from rectangular to polar form.
Note: Plotting the point may be helpful. Exact where possible; otherwise, round to
tenths.
(0, 4)
(–3, 0)
(2, 2)
(–2, 2)
(–3, –3 3 )
(4, –6)

8.
Graphs of functions: Sketch each of the following by yourself. Check with a calculator.
Sketch the set of all points given that:
5
a. r = 3
b.  
c. r   , where 0 ≤  ≤ 
6


9.
a. How would the graph of r = –3 compare to the graph in 8a?
b. Write a rectangular coordinate equation for the graph in 8a.
c. Give another polar equation that will look the same as the graph in 8b.
d. Write a rectangular coordinate equation for the graph in 8b.
e. Sketch the graph of r   for –  ≤  ≤ 


Polar 1.4
Rev. F05