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