Converting Equations from Polar Form to Rectangular Form Sec. 6.4.

Converting EquationsConverting Equationsfrom Polar Formfrom Polar Form

to Rectangular Formto Rectangular FormSec. 6.4Sec. 6.4

First, remind me of the Coordinate Conversion Equations…

x = r cos 0

y = r sin 0

r = x + y2 2 2

tan 0 = yx

Now, on with the Examples…

4secθr

cosθ 4r

4secθ

r

Convert r = 4 sec 0 to rectangular form and identify the graph.Support your answer graphically.

4x

HowHow did I reach each step??? did I reach each step???

A vertical line through x = 4!A vertical line through x = 4! Check your calculator!Check your calculator!

Now, on with the Examples…

4cosθr

2 2 4x y x

2 4 cosθr r

Convert r = 4 cos 0 to rectangular form and identify the graph.Support your answer graphically.

2 24 0x x y A circle with center (2, 0)A circle with center (2, 0)

and radius 2and radius 2

2 22 4x y

2 24 4 4x x y

Conversion EquationsConversion Equations

CTS!!!CTS!!!

Factor!!!Factor!!!

More Examples

2cscθr

Convert the given polar equation to rectangular form. Identifythe graph, and support your answer graphically.

2y horizontal linehorizontal line

More Examples

4cosθr

Convert the given polar equation to rectangular form. Identifythe graph, and support your answer graphically.

2 22 4x y Circle with center (–2, 0) Circle with center (–2, 0)

and radius 2and radius 2

More Examples

secθ 3r

Convert the given polar equation to rectangular form. Identifythe graph, and support your answer graphically.

223 9

2 4x y

Circle with center (3/2, 0) Circle with center (3/2, 0)

and radius 3/2and radius 3/2

More Examples

4cosθ 4sin θr

Convert the given polar equation to rectangular form. Identifythe graph, and support your answer graphically.

2 22 2 8x y Circle with center (2, –2) Circle with center (2, –2)

and radius 2 2and radius 2 2

Converting EquationsConverting Equationsfrom Rectangular Formfrom Rectangular Form

to Polar Formto Polar Form

Right in with an example…

2 22 1 1x x y 2cosθr 2 2 cosθ 0r r

Convert the given rectangular equation to polar form. Identifythe graph, and support your answer graphically.

Circle with center (1, 0)Circle with center (1, 0)and radius 1and radius 1

2cosθ 0r r 2 21 1x y

Conversion EquationsConversion Equations

Expanded the binomial!! Expanded the binomial!!

Guided PracticeConvert the given rectangular equation to polar form. Identifythe graph, and support your answer graphically.

Vertical lineVertical line

8x cos 8r

8

cosr

8sec

Guided PracticeConvert the given rectangular equation to polar form. Identifythe graph, and support your answer graphically.

Line with slope –3/4 and Line with slope –3/4 and yy-intercept 1/2-intercept 1/2

3 4 2x y 3 cos 4 sin 2r r 3cos 4sin 2r

2

3cos 4sinr

Guided PracticeConvert the given rectangular equation to polar form. Identifythe graph, and support your answer graphically.

2 23 2 13x y

2 26 9 4 4 13x x y y 2 2 6 4 0x y x y

2 6 cos 4 sin 0r r r 6cos 4sin 0r r

0r

Guided PracticeConvert the given rectangular equation to polar form. Identifythe graph, and support your answer graphically.

Circle with center (3, 2) and radius 13Circle with center (3, 2) and radius 13

6cos 4sin 0r r

6cos 4sin 0r OR

6cos 4sinr Check the graph?

One more use for polar coordinates…

Radar tracking often gives location in polar coordinates. Supposethat 2 airplanes are at the same altitude with polar coordinatesof (8 mi, 110 ) and (15 mi, 15 ). How far apart are the airplanes?

First, let’s see a graph…

Next, use the Law of Cosines!!!

The airplanes are about 17.604 miles apartThe airplanes are about 17.604 miles apart

2 2 28 15 2 8 15 cos 110 15d

2 28 15 2 8 15 cos95d 17.604d

Another ExampleUsing radar, a navy ship locates two enemy ships at polarcoordinates of (5230m, 130 ) and (6721m, 155 ). Find thedistance between the two enemy ships.

The ships are about 2968.131 meters apartThe ships are about 2968.131 meters apart