do now - #18 on p.558 graph the set of points whose polar coordinates satisfy the given equations...
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Do Now - #18 on p.558Graph the set of points whose polar coordinates satisfy thegiven equations and inequalities.
0 2 1 2r
1 2
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Relating Polar and Cartesian
CoordinatesSection 10.5b
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Relating Polar and Cartesian CoordinatesCoordinate ConversionEquations:
cosx r siny r
2 2 2x y r
tany
x
Ray2
Initial Ray
,P x y ,P r
x
yr
0 Common
originO
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Relating Polar and Cartesian CoordinatesSome curves are easier to work with in polar coordinates,others in Cartesian coordinates… Observe:
cos 2r Polar Equation Cartesian Equivalent
2x 4xy 2 cos sin 4r
2 2 2 2cos sin 1r r 2 2 1x y
1 2 cosr r 2 23 4 1 0y x x
1 cosr 4 4 2 22x y x y 3 2 22 2 0x xy y
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Relating Polar and Cartesian CoordinatesFind a polar equation for the circleSupport graphically.
22 3 9x y
Expand and simplify: 2 2 6 9 9x y y 2 2 6 0x y y
Conversion equations:2 6 sin 0r r
Algebra: 6sin 0r r
0r or 6sinr Check the graph!
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Relating Polar and Cartesian CoordinatesFind a Cartesian equivalent for the polar equation. Identifythe graph.
(a) 2 4 cosr r 2 2 4x y x Conversion equations
2 24 0x x y 2 24 4 4x x y Completing the square
2 22 4x y The graph of the equivalent Cartesian equation is a circlewith radius 2 and center (2, 0).
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Relating Polar and Cartesian CoordinatesFind a Cartesian equivalent for the polar equation. Identifythe graph.
(b)4
2cos sinr
Conversion equations
2cos sin 4r
The graph of the equivalent Cartesian equation is a linewith slope 2 and y-intercept –4.
2 cos sin 4r r 2 4x y
2 4y x
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Exploration 2The polar curves and , wheren is an integer and , are rose curves.
cosr a n1n
2cosr n
sinr a n
1. Graph for . Describethe curves.
2, 4, 6n
Graph window: [–4.7, 4.7] by [–3.1, 3.1]
The graphs are rose curves with 4 petals when , 2n8 petals when , and 12 petals when .4n 6n
2. What is the shortest length a -interval can have and stillproduce the graphs in (1)?
Shortest interval:2
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Exploration 2The polar curves and , wheren is an integer and , are rose curves.
cosr a n1n
2cosr n
sinr a n
3. Based on your observations in (1), describe the graph of when n is a nonzero even integer.
2 nThe graph is a rose curve with petals.
2cosr n4. Graph for . Describethe curves.
3, 5, 7n
Graph window: [–4.7, 4.7] by [–3.1, 3.1]
The graphs are rose curves with 3 petals when , 3n5 petals when , and 7 petals when .5n 7n
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Exploration 2The polar curves and , wheren is an integer and , are rose curves.
cosr a n1n
sinr a n
5. What is the shortest length a -interval can have and stillproduce the graphs in (4)?
Shortest interval:
2cosr n6. Based on your observations in (4), describe the graph of when n is a nonzero odd integer differentfrom .
nThe graph is a rose curve with petals.
1
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Guided PracticeReplace the polar equation by an equivalent Cartesianequation. Then identify or describe the graph.
cot cscr sin cotr
xy
y
2y xA parabola that opens to the right
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Guided PracticeReplace the polar equation by an equivalent Cartesianequation. Then identify or describe the graph.
2 22 cos sin 1r r
The union oftwo lines
2 2 cos sin 1r r r 2 2 2 1x y xy 2 22 1x xy y
2 1x y 1x y
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Guided PracticeReplace the polar equation by an equivalent Cartesianequation. Then identify or describe the graph.
8sinr
A circle with center (0, 4)and radius 4
2 8 sinr r 2 2 8x y y
2 2 8 0x y y 2 2 8 16 16x y y
22 4 16x y
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Guided PracticeReplace the Cartesian equation by an equivalent polarequation. Support graphically.
3x y
How about the graph?
cos sin 3r r
cos sin 3r 3
cos sinr
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Guided PracticeReplace the Cartesian equation by an equivalent polarequation. Support graphically.
2 2 1x xy y
How about the graph?
2 2cos cos sin sin 1r r r r
2 2 2 2cos cos sin sin 1r r r
2 2 2cos cos sin sin 1r
2 1 cos sin 1r Graph: 1
1 cos sinr