Do Now - #18 on p.558Graph the set of points whose polar coordinates satisfy thegiven equations and inequalities.

0 2 1 2r

1 2

Relating Polar and Cartesian

CoordinatesSection 10.5b

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

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

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!

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).

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

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

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

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

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

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

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

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

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