slide 7.4 - 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

Slide 7.4 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Polar Coordinates

Learn vocabulary for polar coordinates.Learn conversion between polar and rectangular coordinates.Learn to convert equations between rectangular and polar forms. Learn to graph polar equations.

SECTION 7.4

1

2

3

4

Slide 7.4 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

POLAR COORDINATES

In a polar coordinate system, we draw a horizontal ray in the plane.

The ray is called the polar axis, and its endpoint is called the pole.

A point P in the plane is described by an ordered pair of numbers (r, ), and we refer to r and as polar coordinates of P.

Slide 7.4 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

POLAR COORDINATES

The point P(r, ) in the polar coordinate system.

r is the “directed distance” from the pole O to the point P. is a directed angle from the polar axis to the line segment OP.

Slide 7.4 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

POLAR COORDINATES

The polar coordinates of a point are not unique. The polar coordinates (3, 60º), (3, 420º), and (3, –300º) all represent the same point.In general, if a point P has polar coordinates (r, ), then for any integer n,

(r, + n • 360º) or (r, + 2nπ)are also polar coordinates of P.

Slide 7.4 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 1 Finding Different Polar Coordinates

Plot the point P with polar coordinates (3, 225º). Find another pair of polar coordinates of P for which the following is true.a. r < 0 and 0º < < 360º b. r < 0 and –360º < < 0ºc. r > 0 and –360º < < 0º

Solution

Slide 7.4 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued

a. r < 0 and 0º < < 360º

EXAMPLE 1 Finding Different Polar Coordinates

Slide 7.4 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued

EXAMPLE 1 Finding Different Polar Coordinates

b. r < 0 and –360º < < 0º

Slide 7.4 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued

EXAMPLE 1 Finding Different Polar Coordinates

c. r > 0 and –360º < < 0º

Slide 7.4 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

POLAR AND RECTANGULAR COORDINATES

Let the positive x-axis of the rectangular coordinate system serve as the polar axis and the origin as the pole for the polar coordinate system.

Each point P has both polar coordinates (r, ) and rectangular coordinates (x, y).

Slide 7.4 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

RELATIONSHIP BETWEENPOLAR AND RECTANGULAR COORDINATES

x2 y2 r2

sin y

r

cos x

r

tan y

x

Slide 7.4 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

CONVERTING FROMPOLAR TO RECTANGULAR COORDINATES

and y r sin.x r cos

To convert the polar coordinates (r, ) of a point to rectangular coordinates, use the equations

Slide 7.4 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 2Convert from Polar to Rectangular Coordinates

Convert the polar coordinates of each point to rectangular coordinates.

b. 4,3

a. 2, 30º

Solutiony r sina. x r cos

x 2 cos 30º x 2 cos 30º

x 23

2

3

y 2sin 30º y 2sin 30º

y 21

2

1

The rectangular coordinates of (2, –30º) are

3, –1 .

Slide 7.4 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 2Convert from Polar to Rectangular Coordinates

b. 4,3

Solution continued

y r sinb. x r cos

x 4 cos3

x 41

2

2

y 4 sin3

y 43

2

2 3

The rectangular coordinates ofare 2, 2 3 .

4,3

Slide 7.4 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

CONVERTING FROM RECTANGULAR TO POLAR COORDINATES

r x2 y2

To convert the rectangular coordinates (x, y) of a point to polar coordinates,

1. Find the quadrant in which the given point (x, y) lies.

2. Use to find r.

tan y

x,3. Find by using and choose so

that it lies in the same quadrant as (x, y).

Slide 7.4 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3Convert from Rectangular to Polar Coordinates

Find polar coordinates (r, ) of the point P whose rectangular coordinates arewith r > 0 and 0 ≤ < 2π.

2,2 3 ,

r x2 y22.

Solution

1. The point P

2,2 3 lies in quadrant II with

x 2 and y 2 3.

r 2 2 2 3 2r 4 12 16 4

Slide 7.4 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3Convert from Rectangular to Polar Coordinates

tan y

x3.

Solution continued

23

Choose because it lies in quadrant II.

The polar coordinates of 2,2 3 are 4,23

.

tan 2 3

2 3

So 3

23

or 2 3

53

Slide 7.4 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EQUATIONS IN RECTANGULAR AND POLAR FORMS

r sin,

An equation that has the rectangular coordinates x and y as variable is called a rectangular (or Cartesian) equation. An equation where the polar coordinates r and are the variables is called a polar equation. Some examples of polar equations are

r 1 cos, and r .

Slide 7.4 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

CONVERTING AN EQUATION FROM RECTANGULAR TO POLAR FORM

To convert a rectangular equation to a polar equation, we simply replace x by r cos and y by r sin , and then simplify where possible.

Slide 7.4 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 5Converting an Equation from Rectangular to Polar Form

Convert the equation x2 + y2 – 3x + 4 = 0 to polar form.

Solution x2 y2 3x 4 0

r cos 2 r sin 2 3 r cos 4 0

r2 cos2 r2 sin2 3r cos 4 0

r2 cos2 sin2 3r cos 4 0

r2 3r cos 4 0

The equationof the rectangular equation x2 + y2 – 3x + 4 = 0.

r2 3r cos 4 0 is the polar form

Slide 7.4 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

CONVERTING AN EQUATION FROM POLAR TO RECTANGULAR FORM

Converting an equation from polar to rectangular form will frequently require some ingenuity in order to use the substitutions

x2 y2 r2 , sin y

r,

cos x

r, and tan

y

x.

Slide 7.4 - 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 6Converting an Equation from Polar to Rectangular Form

Convert each polar equation to a rectangular equation and identify its graph.a. r 3 b. 45º

c. r csc d. r 2 cos

Solutiona. r 3

r2 32

x2 y2 9

Circle: center (0, 0) radius = 3

Slide 7.4 - 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 6Converting an Equation from Polar to Rectangular Form

b. 45ºSolution continued

tan tan 45ºy

x1

Line through the origin with a slope of 1.

y x

Slide 7.4 - 24 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 6Converting an Equation from Polar to Rectangular Form

c. r cscSolution continued

r 1

sinr sin 1

Horizontal line with y-intercept = 1.

y 1

Slide 7.4 - 25 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 6Converting an Equation from Polar to Rectangular Form

d. r 2 cosSolution continued

r2 2r cosx2 y2 2x

Circle: Center (1, 0) radius = 1

x2 2x y2 0x2 2x 1 y2 1

x 1 2 y2 1

Slide 7.4 - 26 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

THE GRAPH OF A POLAR EQUATION

To graph a polar equation we plot points in polar coordinates. The graph of a polar equation

is the set of all points P(r, ) that have at least one polar coordinate representation that satisfies the equation.

r f

Make a table of several ordered pair solutions (r, ) of the equation, plot the points and join them with a smooth curve.

Slide 7.4 - 27 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 7 Sketching the Graph of a Polar Equation

Sketch the graph of the polar equation

Solutioncos (–) = cos , so the graph is symmetric in the polar axis, so compute values for 0 ≤ ≤ π.

r 2 1 cos .

2 1 cos

6

23

3

56

2

r 0

4 2 3

3.733 2 1 02 3

0.27

Slide 7.4 - 28 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued

EXAMPLE 7 Sketching the Graph of a Polar Equation

This type of curve is called a cardiod because it resembles a heart.

r 2 1 cos

Slide 7.4 - 29 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

SYMMETRY IN POLAR EQUATIONS

Symmetry with respect to the polar axis (x-axis)

Replace (r, ) by (r, –)or(–r, π – ).

Slide 7.4 - 30 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

SYMMETRY IN POLAR EQUATIONS

Symmetry with respect to the line (y-axis)

Replace (r, ) by (r, π – )or(–r, –).

2

Slide 7.4 - 31 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

SYMMETRY IN POLAR EQUATIONS

Symmetry with respect to the pole

Replace (r, ) by (r, π + )or(–r, ).

Slide 7.4 - 32 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

LIMAÇONSr a b cos, r a bsin, a 0, b 0

Slide 7.4 - 33 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

LIMAÇONSr a b cos, r a bsin, a 0, b 0

Slide 7.4 - 34 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

ROSE CURVESr a cosn, r asin n, a 0

If n is odd, the rose has n petals.If n is even, the rose has 2n petals

Slide 7.4 - 35 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

ROSE CURVESr a cosn, r asin n, a 0

If n is odd, the rose has n petals.If n is even, the rose has 2n petals

Slide 7.4 - 36 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

CIRCLES

r a cos r asin

Slide 7.4 - 37 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

LEMNISCATES

r2 a2 cos2 r2 a2 sin 2

Slide 7.4 - 38 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

SPIRALS

r a r a

r ak