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)
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
1tany
x
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
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
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
12
Finding dy/dx
• We know r = f(θ) and y = r sin θ and x = r cos θ
• Then
• And
( ) sin ( ) cosy f x f
/
/
dy dy d
dx dx d
13
Finding dy/dx
• Since
• Then
/
/
dy dy d
dx dx d
' sin cos
' cos sin
' sin cos
' cos sin
f fdy
dx f f
r r
r r
14
Example
• Given r = cos 3θ Find the slope of the line tangent at (1/2, π/9)
dy/dx = ?
Evaluate
•3sin3 sin cos3 cos
3sin 3 cos cos3 sin
dy
dx
.160292dy
dx
Define for Calculator
• It is possible to define this derivative as a function on your calculator
15
16
Try This!
• Find where the tangent line is horizontal for r = 2 cos θ
• Find dy/dx
• Set equal to 0, solve for θ