Rectangular to Polar Form for
Equations
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
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Printed: March 23, 2015
AUTHORS
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
www.ck12.org
C HAPTER
Chapter 1. Rectangular to Polar Form for Equations
1
Rectangular to Polar Form
for Equations
Here you’ll learn to convert equations expressed in rectangular coordinates to equations expressed in polar coordinates through substitution.
You are working diligently in your math class when your teacher gives you an equation to graph:
(x + 1)2 − (y + 2)2 = 7
As you start to consider how to rearrange this equation, you are told that the goal of the class is to convert the
equation to polar form instead of rectangular form.
Can you find a way to do this?
By the end of this Concept, you’ll be able to convert this equation to polar form.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/53132
James Sousa Example: Find the Polar Equation for a Line
Guidance
Interestingly, a rectangular coordinate system isn’t the only way to plot values. A polar system can be useful.
However, it will often be the case that there are one or more equations that need to be converted from rectangular to
polar form. To write a rectangular equation in polar form, the conversion equations of x = r cos θ and y = r sin θ are
used.
If the graph of the polar equation is the same as the graph of the rectangular equation, then the conversion has been
determined correctly.
1
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(x − 2)2 + y2 = 4
The rectangular equation (x − 2)2 + y2 = 4 represents a circle with center (2, 0) and a radius of 2 units. The polar
equation r = 4 cos θ is a circle with center (2, 0) and a radius of 2 units.
Example A
Write the rectangular equation x2 + y2 = 2x in polar form.
p
Solution: Remember r = x2 + y2 , r2 = x2 + y2 and x = r cos θ.
2
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Chapter 1. Rectangular to Polar Form for Equations
x2 + y2 = 2x
r2 = 2(r cos θ)
Pythagorean T heorem and x = r cos θ
r2 = 2r cos θ
r = 2 cos θ
Divide each side by r
Example B
Write (x − 2)2 + y2 = 4 in polar form.
Remember x = r cos θ and y = r sin θ.
(x − 2)2 + y2 = 4
(r cos θ − 2)2 + (r sin θ)2 = 4
2
2
2
x = r cos θ and y = r sin θ
2
r cos θ − 4r cos θ + 4 + r sin θ = 4
expand the terms
r2 cos2 θ − 4r cos θ + r2 sin2 θ = 0
subtract 4 f rom each side
2
2
2
2
r cos θ + r sin θ = 4r cos θ
isolate the squared terms
r2 (cos2 θ + sin2 θ) = 4r cos θ
f actor r2 − a common f actor
r2 = 4r cos θ
Pythagorean Identity
r = 4 cos θ
Divide each side by r
Example C
Write the rectangular equation (x + 4)2 + (y − 1)2 = 17 in polar form.
(x + 4)2 + (y − 1)2 = 17
(r cos θ + 4)2 + (r sin θ − 1)2 = 17
2
2
2
2
2
2
2
x = r cos θ and y = r sin θ
2
r cos θ + 8r cos θ + 16 + r sin θ − 2r sin θ + 1 = 17
2
2
r cos θ + 8r cos θ − 2r sin θ + r sin θ = 0
2
2
r cos θ + r sin θ = −8r cos θ + 2r sin θ
2
2
2
expand the terms
subtract 17 f rom each side
isolate the squared terms
r (cos θ + sin θ) = −2r(4 cos θ − sin θ)
f actor r2 − a common f actor
r2 = −2r(4 cos θ − sin θ)
Pythagorean Identity
r = −2(4 cos θ − sin θ)
Divide each side by r
Guided Practice
1. Write the rectangular equation (x − 4)2 + (y − 3)2 = 25 in polar form.
2. Write the rectangular equation 3x − 2y = 1 in polar form.
3. Write the rectangular equation x2 + y2 − 4x + 2y = 0 in polar form.
Solutions:
1.
3
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(x − 4)2 + (y − 3)2 = 25
x2 − 8x + 16 + y2 − 6y + 9 = 25
x2 − 8x + y2 − 6y + 25 = 25
x2 − 8x + y2 − 6y = 0
x2 + y2 − 8x − 6y = 0
r2 − 8(r cos θ) − 6(r sin θ) = 0
r2 − 8r cos θ − 6r sin θ = 0
r(r − 8 cos θ − 6 sin θ) = 0
r = 0 or r − 8 cos θ − 6 sin θ = 0
r = 0 or r = 8 cos θ + 6 sin θ
From graphing r − 8 cos θ − 6 sin θ = 0, we see that the additional solutions are 0 and 8.
2.
3x − 2y = 1
3r cos θ − 2r sin θ = 1
r(3 cos θ − 2 sin θ) = 1
r=
1
3 cos θ − 2 sin θ
3.
x2 + y2 − 4x + 2y = 0
r2 cos2 θ + r2 sin2 θ − 4r cos θ + 2r sin θ = 0
r2 (sin2 θ + cos2 θ) − 4r cos θ + 2r sin θ = 0
r(r − 4 cos θ + 2 sin θ) = 0
r = 0 or r − 4 cos θ + 2 sin θ = 0
r = 0 or r = 4 cos θ − 2 sin θ
Concept Problem Solution
The original equation to convert is:
(x + 1)2 − (y + 2)2 = 7
You can substitute x = r cos θ and y = r sin θ into the equation, and then simplify:
(r cos θ + 1)2 − (r sin θ + 2)2 = 7
(r2 cos2 θ + 2r cos θ + 1) − (r2 sin2 θ + 4r sin θ + 4) = 7
r2 (cos2 θ − sin2 θ) + 2r(cos θ − 2 sin θ) − 3 = 7
r2 (cos2 θ − sin2 θ) + 2r(cos θ − 2 sin θ) = 10
4
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Chapter 1. Rectangular to Polar Form for Equations
Explore More
Write each rectangular equation in polar form.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
x=3
y=4
x2 + y2 = 4
x2 + y2 = 9
(x − 1)2 + y2 = 1
(x − 2)2 + (y − 3)2 = 13
(x − 1)2 + (y − 3)2 = 10
(x + 2)2 + (y + 2)2 = 8
(x + 5)2 + (y − 1)2 = 26
x2 + (y − 6)2 = 36
x2 + (y + 2)2 = 4
2x + 5y = 11
4x − 7y = 10
x + 5y = 8
3x − 4y = 15
5