Polar Coordinates Lesson 6.3. Points on a Plane Rectangular coordinate system Represent a point…

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Polar Coordinates Lesson 6.3

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Plot Given Polar Coordinates Locate the following

Transcript of Polar Coordinates Lesson 6.3. Points on a Plane Rectangular coordinate system Represent a point…

Page 1: Polar Coordinates Lesson 6.3. Points on a Plane Rectangular coordinate system  Represent a point…

Polar Coordinates

Lesson 6.3

Page 2: Polar Coordinates Lesson 6.3. Points on a Plane Rectangular coordinate system  Represent a point…

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

r

θ

(x, y)

(r, θ)

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Plot Given Polar Coordinates

• Locate the following

2,4

A

33,2

C

24,3

B

51,4

D

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Find Polar Coordinates

• What are the coordinates for the given points?

• B• A

• C• D

• A =

• B =

• C =

• D =

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Converting Polar to Rectangular

• Given polar coordinates (r, θ) Change to rectangular

• By trigonometry x = r cos θ

y = r sin θ

• Try = ( ___, ___ )

θ

r

x

y

2,4

A

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Converting Rectangular to Polar

• Given a point (x, y) Convert to (r, θ)

• By Pythagorean theorem r2 = x2 + y2

• By trigonometry

• Try this one … for (2, 1) r = ______ θ = ______

θ

r

x

y

1tan yx

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Polar Equations

• States a relationship between all the points (r, θ) that satisfy the equation

• Example r = 4 sin θ Resulting values

θ in degrees

Note: for (r, θ)

It is θ (the 2nd element that is the independent

variable

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Graphing Polar Equations

• Set Mode on TI calculator Mode, then Graph => Polar

• Note difference of Y= screen

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Graphing Polar Equations

• Also best to keepangles in radians

• Enter function in Y= screen

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Graphing Polar Equations

• Set Zoom to Standard,

then Square

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Try These!

• For r = A cos B θ Try to determine what affect A and B have

• r = 3 sin 2θ• r = 4 cos 3θ• r = 2 + 5 sin 4θ

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Polar Form Curves

• Limaçons r = B ± A cos θ r = B ± A sin θ

3 5cosr

3 2sinr

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Polar Form Curves

• Cardiods Limaçons in which a = b r = a (1 ± cos θ) r = a (1 ± sin θ)

3 3sinr

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Polar Form Curves

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

5cos3r

5sin 4r

a

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Polar Form Curves

• Lemiscates r2 = a2 cos 2θ r2 = a2 sin 2θ

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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

4cos4sin

rr

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Assignment A

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

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Area of a Sector of a Circle

• Given a circle with radius = r Sector of the circle with angle = θ

• The area of the sector given by

θr

212

A r

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Area of a Sector of a Region

• Consider a region bounded by r = f(θ)

• A small portion (a sector with angle dθ) has area

dθ α

β

21 ( )2

A f d

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Area of a Sector of a Region

• We use an integral to sum the small pie slices

α

β

2

2

1 ( )2

12

A f d

r d

r = f(θ)

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Guidelines

1. Use the calculator to graph the region• Find smallest value θ = a, and largest value

θ = b for the points (r, θ) in the region

2. Sketch a typical circular sector• Label central angle dθ

3. Express the area of the sector as4. Integrate the expression over the limits

from a to b

212

A r

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Find the Area• Given r = 4 + sin θ

Find the area of the region enclosed by the ellipse

2

2

0

1 4 sin2

d

The ellipse is traced out by

0 < θ < 2π

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Areas of Portions of a Region

• Given r = 4 sin θ and rays θ = 0, θ = π/3

/32

0

1 16sin2

d

The angle of the rays specifies the limits of

the integration

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Area of a Single Loop

• Consider r = sin 6θ Note 12 petals θ goes from 0 to 2π One loop goes from

0 to π/6

/ 6

2

0

1 sin 62

d

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Area Of Intersection

• Note the area that is inside r = 2 sin θand outside r = 1

• Find intersections• Consider sector for a dθ

Must subtract two sectors

56 6

and

5 / 6

2 2

/ 6

1 2sin 12

d

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Assignment B

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