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Plot Given Polar Coordinates Locate the following
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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
• Also possible to represent different ways• Consider using dist from origin, angle formed
with positive x-axis
•
•
r
θ
(x, y)
(r, θ)
Plot Given Polar Coordinates
• Locate the following
2,4
A
33,2
C
24,3
B
51,4
D
Find Polar Coordinates
• What are the coordinates for the given points?
• B• A
• C• D
• A =
• B =
• C =
• D =
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
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
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
Graphing Polar Equations
• Set Mode on TI calculator Mode, then Graph => Polar
• Note difference of Y= screen
Graphing Polar Equations
• Also best to keepangles 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
3 5cosr
3 2sinr
Polar Form Curves
• Cardiods Limaçons in which a = b r = a (1 ± cos θ) r = a (1 ± sin θ)
3 3sinr
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
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
• Find all intersections
4cos4sin
rr
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
• The area of the sector given by
θr
212
A r
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
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)
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
Find the Area• Given r = 4 + sin θ
Find the area of the region enclosed by the ellipse
dθ
2
2
0
1 4 sin2
d
The ellipse is traced out by
0 < θ < 2π
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
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
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
dθ
56 6
and
5 / 6
2 2
/ 6
1 2sin 12
d
Assignment B
• Lesson 6.3 B• Page 384• Exercises 31 – 53 odd