Intro to Polar Coordinates

Lesson 6.5A

2

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)

3

Plot Given Polar Coordinates

Locate the following

2,4

A

33,2

C

24,3

B

51,

4D

4

Find Polar Coordinates

What are the coordinates for the given points?

• B• A

• C

• D

• A =

• B =

• C =

• D =

5

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

6

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

7

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

8

Graphing Polar Equations

Set Mode on TI calculator Mode, then Graph => Polar

Note difference of Y= screen

9

Graphing Polar Equations

Also best to keepangles in radians

Enter function in Y= screen

10

Graphing Polar Equations

Set Zoom to Standard,

then Square

11

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

Experiment with Polar Function Spreadsheet

Experiment with Polar Function Spreadsheet

12

Assignment A

Lesson 6.5A Page 424 Exercises 1 – 47 odd

Polar Coordinates

Lesson 6.5B

14

Write Polar Equation in Rectangular Form

Given r = 2 sin θ Write as rectangular

equation Use definitions

And identities(see inside back cover)

Graph the given equation for clues

1

2 2 2

cos

sin

tan

x r

y r

y

x

r x y

1

2 2 2

cos

sin

tan

x r

y r

y

x

r x y

15

Write Polar Equation in Rectangular Form

Given r = 2 sin θ

We know

Thus

And

2

2 2

sin2

2

2

r y

r

r y

x y y

16

Write Rectangular Equation in Polar Form

Consider 2x – 3y = 6

As before, usedefinitions

1

2 2 2

cos

sin

tan

x r

y r

y

x

r x y

1

2 2 2

cos

sin

tan

x r

y r

y

x

r x y

2 cos 3 sin 6

2cos 3sin 6

6

2cos 3sin

r r

r

r

17

Assignment B

Lesson 6.5B Page 424 Exercises 49 – 73 odd