Polar Coordinates

1
Polar Coordinates

• Lesson 6.3

2
Points on a Plane

• Rectangular coordinate system
• Represent a point by two distances from the
  origin
• Horizontal dist, Vertical dist
• Also possible to represent different ways
• Consider using dist from origin, angle formed
  with positive x-axis

(x, y)

(r, ?)
3
Plot Given Polar Coordinates

• Locate the following

4
Find Polar Coordinates

• What are the coordinates for the given points?

A

• A
• B
• C
• D

B
D
C
5
Converting Polar to Rectangular

• Given polar coordinates (r, ?)
• Change to rectangular
• By trigonometry
• x r cos ?y r sin ?
• Try ( ___, ___ )

r
y
?
x
6
Converting Rectangular to Polar

• Given a point (x, y)
• Convert to (r, ?)
• By Pythagorean theorem r2 x2 y2
• By trigonometry
• Try this one for (2, 1)
• r ______
• ? ______

r
y
?
x
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Polar Equations

• States a relationship between all the points (r,
  ?) that satisfy the equation
• Example r 4 sin ?
• Resulting values

Note for (r, ?) It is ? (the 2nd element that
is the independent variable
? in degrees
8
Graphing Polar Equations

• Set Mode on TI calculator
• Mode, then Graph gt Polar
• Note difference of Y screen

9
Graphing Polar Equations

• Also best to keepangles in radians
• Enter function in Y screen

10
Graphing Polar Equations

• Set Zoom to Standard,
• then Square

11
Try These!

• For r A cos B ?
• Try to determine what affect A and B have
• r 3 sin 2?
• r 4 cos 3?
• r 2 5 sin 4?

12
Polar Form Curves

• Limaçons
• r B A cos ?
• r B A sin ?

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Polar Form Curves

• Cardiods
• Limaçons in which a b
• r a (1 cos ?)
• r a (1 sin ?)

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Polar Form Curves
a

• Rose Curves
• r a cos (n ?)
• r a sin (n ?)
• If n is odd ? n petals
• If n is even ? 2n petals

15
Polar Form Curves

• Lemiscates
• r2 a2 cos 2?
• r2 a2 sin 2?

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Intersection of Polar Curves

• Use all tools at your disposal
• Find simultaneous solutions of given systems of
  equations
• Symbolically
• Use Solve( ) on calculator
• Determine whether the pole (the origin) lies on
  the two graphs
• Graph the curves to look for other points of
  intersection

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Finding Intersections

• Given
• Find all intersections

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Assignment A

• Lesson 6.3A
• Page 384
• Exercises 3 29 odd

19
Area of a Sector of a Circle

• Given a circle with radius r
• Sector of the circle with angle ?
• The area of the sector given by

?
r
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Area of a Sector of a Region

• Consider a region bounded by r f(?)
• A small portion (a sector with angle d?) has area

ß

d?
a

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Area of a Sector of a Region

• We use an integral to sum the small pie slices

ß

r f(?)
a

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Guidelines

• Use the calculator to graph the region
• Find smallest value ? a, and largest value ?
  b for the points (r, ?) in the region
• Sketch a typical circular sector
• Label central angle d?
• Express the area of the sector as
• Integrate the expression over the limits from a
  to b

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Find the Area

• Given r 4 sin ?
• Find the area of the region enclosed by the
  ellipse

d?
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Areas of Portions of a Region

• Given r 4 sin ? and rays ? 0, ? p/3

The angle of the rays specifies the limits of the
integration
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Area of a Single Loop

• Consider r sin 6?
• Note 12 petals
• ? goes from 0 to 2p
• One loop goes from0 to p/6

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Area Of Intersection

• Note the area that is inside r 2 sin ?and
  outside r 1
• Find intersections
• Consider sector for a d?
• Must subtract two sectors

d?
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Assignment B

• Lesson 6.3 B
• Page 384
• Exercises 31 53 odd