Transcript Polar Coordinates

Polar Coordinates
Lesson 10.5
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
Finding dy/dx
• We know
 r = f(θ) and y = r sin θ and x = r cos θ
• Then
y  f ( )  sin 
• And
dy dy / d

dx dx / d
x  f ( )  cos 
12
Finding dy/dx
• Since
dy dy / d

dx dx / d
• Then
dy f '    sin   f    cos 

dx f '    cos   f    sin 
r ' sin   r  cos 

r ' cos   r  sin 
13
Example
• Given r = cos 3θ
 Find the slope of the line tangent at (1/2, π/9)
 dy/dx = ?
dy 3sin 3  sin   cos 3  cos 

dx 3sin 3  cos   cos 3  sin 
 Evaluate
•
dy
 .160292
dx
14
Define for Calculator
• It is possible to define this derivative as a
function on your calculator
15
Try This!
• Find where the tangent line
is horizontal for r = 2 cos θ
• Find dy/dx
• Set equal to 0, solve for θ
16
Assignment
• Lesson 10.4
• Page 736
• Exercises 1 – 19 odd,
23 – 26 all
• Exercises 69 – 91 EOO