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

• 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,

4D

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

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

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

Assignment

• Lesson 10.4

• Page 736

• Exercises 1 – 19 odd, 23 – 26 all

• Exercises 69 – 91 EOO