Using Polar Coordinates

1
Using Polar Coordinates

• Graphing and converting polar and rectangular
  coordinates

2
Graphing Polar Coordinates
The grid at the left is a polar grid. The
typical angles of 30o, 45o, 90o, are shown on
the graph along with circles of radius 1, 2, 3,
4, and 5 units.
A
Points in polar form are given as (r, ß ) where r
is the radius to the point and ß is the angle of
the point.
On one of your polar graphs, plot the point (3,
90o)?
The point on the graph labeled A is correct.
3
Graphing Polar Coordinates
Now, try graphing .
C
A
Did you get point B?
Polar points have a new aspect. A radius can be
negative! A negative radius means to go in the
exact opposite direction of the angle.
B
To graph (-4, 240o), find 240o and move 4 units
in the opposite direction. The opposite
direction is always a 180o difference.
Point C is at (-4, 240o). This point could also
be labeled as (4, 60o).
4
Graphing Polar Coordinates
How would you write point A with a negative
radius?
C
A
A correct answer would be (-3, 270o) or (-3,
-90o).
In fact, there are an infinite number of ways to
label a single polar point. Is (3, 450o) the
same point?
B
Dont forget, you can also use radian angles as
well as angles in degrees.
On your own, find at least 4 different polar
coordinates for point B.
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Graphing Polar Coordinates
On your own, find at least 4 different polar
coordinates for point B.
C
A
There are many possible answers.Here are just a
few.(2, 225o), (2, -135o)(-2, 45o), (-2,
-315o)One could add or subtract 360o to any
ofthe above angles.Radians would result in(2,
5p/4), (2, -3p/4)(-2, p/4), (-2, -7p/4)One
could add or subtract 2p to any previous radian
angle.
B
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Converting from Rectangular to Polar
Find the polar form for the rectangular point (4,
3).
To find the polar coordinate, we must calculate
the radius and angle to the given point.
(4, 3)
r
We can use our knowledge of right triangle
trigonometry to find the radius and angle.
3
ß
4
r2 32 42r2 25r 5
tan ß ¾ ß tan-1(¾) ß 36.87o or 0.64 rad
The polar form of the rectangular point (4, 3)
is (5, 36.87o)
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Converting from Rectangular to Polar
In general, the rectangular point (x, y) is
converted to polar form (r, ?) by
1. Finding the radius
r2 x2 y2
(x, y)
2. Finding the angle
r
y
ß
tan ß y/x or ß tan-1(y/x)Recall that some
angles require the angle to be converted to the
appropriate quadrant.
x
Note This is just like finding the length and
direction angle of a vector!
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Converting from Rectangular to Polar
On your own, find polar form for the point (-2,
3).
(-2, 3)
r2 (-2)2 32r2 4 9r2 13r
However, the angle must be in the second
quadrant, so we add 180o to the answer and get an
angle of 123.70o. The polar form is ( ,
123.70o)
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Converting from Polar to Rectanglar
Convert the polar point (4, 30o) to rectangular
coordinates.
We are given the radius of 4 and angle of 30o.
Find the values of x and y.
Using trig to find the values of x and y, we know
that cos ß x/r or x r cos ß. Also, sin ß
y/r ory r sin ß.
4
y
30o
x
The point in rectangular form is
10
Converting from Polar to Rectanglar
On your own, convert (3, 5p/3) to rectangular
coordinates.
We are given the radius of 3 and angle of 5p/3 or
300o. Find the values of x and y.
-60o
The point in rectangular form is
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Rectangular and Polar Equations
Equations in rectangular form use variables (x,
y), whileequations in polar form use variables
(r, ß) where ß is an angle.
Converting from one form to another involves
changing the variables from one form to the other.
We have already used all of the conversions which
are necessary.
Converting Polar to Rectangular
Converting Rectanglar to Polar
cos ß x/rsin ß y/rtan ß y/xr2 x2 y2
x r cos ß y r sin ß x2 y2 r2
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Convert Rectangular Equationsto Polar Equations
The goal is to change all xs and ys to rs and
ßs.When possible, solve for r.
Example 1 Convert x2 y2 16 to polar form.
Since x2 y2 r2, substitute into the equation.
r2 16
Simplify.
r 4
r 4 is the equivalent polar equation to x2 y2
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Convert Rectangular Equationsto Polar Equations
Example 2 Convert y 3 to polar form.
Since y r sin ß, substitute into the equation.
r sin ß 3
Solve for r when possible.
r 3 / sin ß
r 3 csc ß is the equivalent polar equation.
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Convert Rectangular Equationsto Polar Equations
Example 3 Convert (x - 3)2 (y 3)2 18 to
polar form.
Square each binomial.
x2 6x 9 y2 6y 9 18
Since x2 y2 r2, re-write and simplify by
combining like terms.
x2 y2 6x 6y 0
Substitute r2 for x2 y2, r cos ß for x and r
sin ß for y.
r2 6rcos ß 6rsin ß 0
Factor r as a common factor.
r(r 6cos ß 6sin ß) 0
r 0 or r 6cos ß 6sin ß 0
Solve for r r 0 or r 6cos ß 6sin ß
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Convert Polar Equationsto Rectangular Equations
The goal is to change all rs and ßs to xs and
ys.
Example 1 Convert r 4 to rectangular form.
Since r2 x2 y2, square both sides to get r2.
r2 16
Substitute.
x2 y2 16
x2 y2 16 is the equivalent polar equation to
r 4
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Convert Polar Equationsto Rectangular Equations
Example 2 Convert r 5 cos ß to rectangular
form.
Since cos ß x/r, substitute for cos ß.
Multiply both sides by r.
r2 5x
Substitute for r2.
x2 y2 5x is rectangular form.
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Convert Polar Equationsto Rectangular Equations
Example 3 Convert r 3 csc ß to rectangular
form.
Since csc ß r/y, substitute for csc ß.
Multiply both sides by y/r.
Simplify
y 3 is rectangular form.
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Assignment
6.4 / 1, 2, 7-13, 15, 16, 19, 20, 27, 28, 31-40