slide 7.4 - 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley
TRANSCRIPT
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
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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.
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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
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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
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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
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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
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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
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SYMMETRY IN POLAR EQUATIONS
Symmetry with respect to the polar axis (x-axis)
Replace (r, ) by (r, –)or(–r, π – ).
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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, ).
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LIMAÇONSr a b cos, r a bsin, a 0, b 0
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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
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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
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CIRCLES
r a cos r asin
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LEMNISCATES
r2 a2 cos2 r2 a2 sin 2
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SPIRALS
r a r a
r ak