Transcript Polar Coordinates

Polar Coordinates
Lesson 6.3
Points on a Plane
• Rectangular coordinate system
 Represent a point by two distances from the
origin
 Horizontal dist, Vertical dist
(x, y)
•
• Also possible to represent different ways
• Consider using dist from origin, angle formed
(r, θ)
with positive x-axis
•
θ
r
Plot Given Polar Coordinates
• Locate the following
 
A   2, 
 4
 2 
B   4,

 3 
3 

C   3, 
2 

5 

D   1, 

4 

Find Polar Coordinates
• What are the coordinates for the given
points?
• B
• A
•D
•C
•A=
•B=
•C=
•D=
Converting Polar to Rectangular
• Given polar coordinates (r, θ)
 Change to rectangular
r
θ
x
• By trigonometry
 x = r cos θ
y = r sin θ
• Try
 
A   2, 
 4
=
( ___, ___ )
•
y
Converting Rectangular to Polar
• Given a point (x, y)
r
 Convert to (r, θ)
θ
• By Pythagorean theorem r2 = x2 +
• By trigonometry
  tan
• Try this one … for (2, 1)
 r = ______
 θ = ______
1
y
x
x
y2
•
y
Polar Equations
• States a relationship between all the points
(r, θ) that satisfy the equation
• Example
r = 4 sin θ
 Resulting values
Note: for (r, θ)
θ in
degrees
It is θ (the 2nd element
that is the independent
variable
Graphing Polar Equations
• Set Mode on TI calculator
 Mode, then Graph => Polar
• Note difference of Y= screen
Graphing Polar Equations
• Also best to keep
angles in radians
• Enter function in
Y= screen
Graphing Polar Equations
• Set Zoom to Standard,
 then Square
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
Polar Form Curves
• Limaçons
 r = B ± A cos θ
 r = B ± A sin θ
r  3  5cos
r  3  2sin 
Polar Form Curves
• Cardiods
 Limaçons in which a = b
 r = a (1 ± cos θ)
 r = a (1 ± sin θ)
r  3  3sin 
Polar Form Curves
• Rose Curves
a
 r = a cos (n θ)
 r = a sin (n θ)
 If n is odd → n petals
 If n is even → 2n petals
r  5cos3
r  5sin 4
Polar Form Curves
• Lemiscates
 r2 = a2 cos 2θ
 r2 = a2 sin 2θ
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
Finding Intersections
• Given
r  4 cos 
r  4sin 
• Find all intersections
Assignment A
• Lesson 6.3A
• Page 384
• Exercises 3 – 29 odd
Area of a Sector of a Circle
• Given a circle with radius = r
 Sector of the circle with angle = θ
θ
r
1 2
• The area of the sector given by A  2  r
Area of a Sector of a Region
• Consider a region bounded by r = f(θ)
•
β
dθ
•
α
• A small portion (a sector with angle dθ) has
1
2
area
A   f ( ) d
2
Area of a Sector of a Region
• We use an integral to sum the small pie
slices
β
r = f(θ)
•
•

1
2
A    f ( )  d
2

1 2
  r d
2
α
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
1 2
A  r
2
3. Express the area of the sector as
4. Integrate the expression over the limits from
a to b
Find the Area
• Given r = 4 + sin θ
 Find the area of the region enclosed by the
ellipse
dθ
The ellipse is
traced out by
0 < θ < 2π
1
2
2
  4  sin  
0
2
d
Areas of Portions of a Region
• Given r = 4 sin θ and rays θ = 0, θ = π/3
The angle of the rays
specifies the limits of
the integration
1
2
 /3
 16sin
0
2
 d
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
1
2
 /6

0
2
sin
 6  d
Area Of Intersection
• Note the area that is
inside r = 2 sin θ
and outside r = 1

dθ
• Find intersections 6
• Consider sector for a dθ
and
5
6
 Must subtract two sectors
1
2
5 / 6

/6
 2sin  2  12  d


Assignment B
• Lesson 6.3 B
• Page 384
• Exercises 31 – 53 odd