Polar Functions
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polar coordinate system
In the rectangular coordinate system, we identify points in the plane by their xand y-coordinates. We can also identify points by using their distance from the origin
and their angle from the positive x-axis. This is called the polar coordinate system.
Begin by drawing a half-line (or ray) from a fixed point called the pole (think
origin). The half-line is called the polar axis (think positive x-axis). Let P be any point in
the plane. Draw a line segment from the pole to point P. This line segment has some
distance r and makes an angle " with the polar axis. Label point P (r," ) .
However, the angle " is not unique for P, e.g., P (r," + 2#k) would also identify
the same point P. Negative values of r are possible, and r is referred to as a directed
distance. E.g., we could identify our point P as ("r,# + $ ) . Since the angle choice
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depends on whether r is positive or negative, the angle is!referred to as a directed angle.
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Thus, a point in the plane may have an infinite!number of polar coordinates.
"
7#
#
Plot the points P (2, ) , Q (0, " )!
, R (1," ) , and S ("2, ) .
4
6
6
Converting between polar and rectangular coordinates
See figure on p. 577. Sometimes we want to convert from polar to rectangular
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coordinates or vice versa.
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To convert from polar to rectangular coordinates,
given a point (r," ) use
x = r cos" or y = r sin " .
To convert from rectangular to polar coordinates, given a point (x, y) use
y
x 2 + y 2 = r 2 or tan " = .
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x
!
!
"
#
3
1. Convert (2, ) to rectangular coordinates. x = r cos" = 2cos = 2( ) = 3 .
6
6
2
!
#
1
!
y = r sin " = 2sin
= 2( ) = 1. So (x, y) = ( 3,1) .
6
2
2
2
2
2
2.!Convert (1, -1) to polar coordinates.
! r = x + y = (1) + ("1) = 2 .
#1
tan " =
= #1 and " = tan#1 (#1)!, since the point (x, y) is in quadrant IV and the arctan is
1
$
#
only defined for quadrants I and IV the!angle " = # . So (r," ) = ( 2," ) . Other
4
4
3#
7"
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possibilities are (" 2, ) and ( 2, ) .
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4
distance formula for polar coordinates !
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See figure 6 on p. 579. Let's!find the distance between
two points in the polar
coordinate system. We can make a triangle by drawing the line segment connecting the
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! the Law of Cosines, we can find the distance between the two
two points.
Then by using
points. d 2 = r12 + r22 " 2r1r2 cos(#2 " #1 ) .
Find the distance between the points (4, " ) and (3,5 " 3) .
d = 4 2 + 32 " 2(4)(3) cos(5 # 3" # ) = 16 + 9 " 24 cos(2 # 3) = 25 " 24("1 2) = 37
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polar functions
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!coordinate system are usually written as
Recall that functions in the rectangular
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y = f(x). Functions in the polar coordinate system are written as r = f( " ), i.e., the
directed distance of a point is a function of the angle " . Examples of polar functions are:
r = " , r = 1, r = sin " , r = 1 + sin " , r = 4/(1 + sin " ), r = d/cos( " – " ).
graphs of polar functions
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Where graphs of rectangular functions are drawn from left to right on the plane,
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graphs !
of polar functions are drawn counterclockwise from the polar axis.
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DRAW the graph of r = sin " .
Let’s begin by making a table of values.
"
0 " /6 " /3 " /2 2 " /3 5 " /6 " 7 " /6 4 " /3 3 " /2 5 " /3 11 " /6 2 "
0 1 2! 3 2 1
3 2 1 2 0 "1 2 " 3 2 "1 " 3 2 "1 2 0
r
Plot these points on the polar coordinate system and sketch a smooth curve
Notice
the graph
is drawn
increases from 0
! connecting
! the
! points.
!
!
!
! !
! counterclockwise
!
!
!as " !
"
"
"
to 2 " !radians.
Notice
that
r
was
negative
when
<
<
2
.
What
effect
! !
! !
! did this have
!
! !
! ! !
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on the graph? It retraced its graph.
When did the graph pass through the pole (origin)? At (0, 0), (0, " ) and (0, 2 " ).
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The graph passes through the pole (origin) when r = 0.
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Sometime it is helpful to determine the values of " for which r is a maximum.
#
Looking at our table of values, for which values of " is r a maximum?
" =! . What is
!
2
"
the maximum value of r? (maximum value of !
r = sin = 1) Could we have predicted
2
!
#
this from the function? Yes, sin " is a maximum when " = !
. What if our function had
2
3#
been r = 1 - 2sin " ? Then r would have
. What is this
! a maximum value when " =
2
!
3#
maximum value of r? (maximum value of r =!
1" 2sin
= 1" 2("1) = 3 )
2
!polar equations of circles
! of a circle. See figure 7 p.
We can use the distance formula to find an equation
580. If (r," ) is an arbitrary point on the circle of radius a with center (r0 ,"0 ) , then
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a 2 = r 2 + r02 " 2rr0 cos(# " #0 ) is an equation of this circle. If the center of the circle is the
origin, then r0 = 0 and a 2 = r 2 (or r = ±a ).
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"
2
Find a polar equation of the circle with radius 1 and center (2, ) .
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$
$
$
12 = r 2!+ 2 2 " 2(2)r!cos(# " ) % 1 = r 2 + 4 " 4r cos(# " ) % r 2 " 4r cos(# " ) = "3
2
2
2
polar equations of lines
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Let's consider polar equations of lines. We have two cases:
1. The line passes through the pole (origin)
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2. The line does not pass through the pole
If a line passes through the pole and makes an angle "0 with the positive x-axis,
then the polar equation of the line is " = "0 . Since r is not related to " , r can assume any
$
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value (both positive and negative). Draw the line " = # .
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See fig. 10 p. 581. For a line that does not pass through the pole, we draw a
perpendicular line segment from the pole to the line L at some point N. Let's say the
polar coordinates of N are (d," ) . Let point P, (r," ) , be an arbitrary point on the line L.
Then in the right triangle ONP, we have cos(" # $ ) = d r or d = r cos(" # $ ) .
$
!4
IF r cos(" # ) =1 is the polar equation of a line, then
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# &
! %1, " (
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1. Find the polar coordinates
of the point N.
$ 4'
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2. Find the polar coordinates of the point on the line where " =
3. Sketch the line.
"&
# #
. % 2, (
2 $
2'
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converting polar functions to rectangular and vice
! versa
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Some graphs are more easily written in polar form than rectangular
form, r = e" .
Other graphs are more easily written in rectangular form, e.g., y = 2x + 1. Sometimes in
calculus we want to convert from one form to another to simplify our calculations. To
convert functions, we use the same four formulas we saw earlier:
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y
y = r sin "
x = r cos"
x 2 + y 2 = r 2 tan " = .
x
Convert x 2 + y 2 = 16 to polar form. Substitute for x and y.
(r cos " ) 2 + (r sin " ) 2 = r 2 (cos 2 " + sin 2 " ) = r 2 (1) = 16 ; r = 4. This is the graph of a circle
! centered at the
! 4.
! pole with radius
!
2
to rectangular form. Converting from polar to
!Convert r =
1" sin #
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rectangular form is not as easy. Multiply both sides by (1 – sin " ). r " r sin # = 2 . Since r
sin " = y, we have r – y = 2 or r = y + 2. Squaring both sides, r 2 = (y + 2) 2 = y 2 + 4 y + 4 .
Substituting for r 2 , we have x 2 + y 2 = y 2 + 4 y + 4 . Subtracting y 2 from both sides, we
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have x 2 = 4 y + 4 = 4(y + 1) . This is the graph of a parabola
! ! with vertex (0, -1).
symmetry tests for polar coordinates
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coordinates may be symmetric about the x-axis, the y-axis, or the
!Graphs in polar
!
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origin like graphs in rectangular coordinates.
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A polar graph is symmetric across the x-axis if r = f (" ) = f (#" ) . This is similar
to our earlier definition of even functions that are symmetric across the x-axis. This
definition makes sense, since the points ( r," ) and ( r,"# ) are reflections of each other
across the x-axis.
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A polar graph is symmetric across the y-axis if f ("# ) = "r or equivalently
f (" # $ ) = r . A polar graph may only satisfy one of these tests. Read Example 3 on p.
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588.
A polar graph is symmetric across the origin if f (" ) = r and f (" ) = #r .
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