Polar Coordinates

• Butterflies are among the most celebrated of all insects.

• Their symmetry can be explored with trigonometric functions and a system for plotting points called the polar coordinate system.

Objective:

Plot each point given in polar coordinates.

The foundation of the polar coordinate system is a horizontal ray that extends to the right. The ray is called the polar axis. The endpoint of the ray is called the pole.

r can be positive, negative, or zero.

Positive angles are measured counterclockwise from the polar axis.

Negative angles are measured clockwise from the polar axis.

• Plot the points with the following polar coordinates:

0(2,135 )

3( 3, )

2

( 1, )4

Objective:

Polar coordinates of a point are given. Find the rectangular coordinates of each point.

• If P is a point with polar coordinates , the rectangular coordinates of P are given by

( , )r

( , )x y

cos

sin

x r

y r

• Find the rectangular coordinates of the point with the following polar coordinates:

(6, )6

• Find the rectangular coordinates of the point with the following polar coordinates:

( 4, )4

Objective:

The rectangular coordinates of a point are given. Find polar coordinates for each point.

1. To find r, compute the distance from the origin to (x, y).

2. To find , first determine the quadrant that the point lies in.

1

1

1

1

: tan

: tan

: tan

: tan

yI

xy

IIxy

IIIx

yIV

x

2 2 2r x y

• Find polar coordinates of a point whose rectangular coordinates are:

(2, 2)

Find the polar form of the rectangular point (4, 3).

Find the polar form for the point (-2, 3).

Find the polar coordinates for the rectangular coordinates (-3, 3).

Objective:

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates.

• Transform the equation from rectangular coordinates to polar coordinates.

4 9xy

• Transform the following equations from rectangular coordinates to polar coordinates.

2 2

2

2 3 5

5

3 4 2

( 3) 9

x

x y

x

x y

x y

Objective:

The letters represent polar coordinates. Write each equation using rectangular coordinates.

Review from Chapter 4Completing the Square

2 12 23 0x x

• Transform the equation from polar coordinates to rectangular coordinates.

4sinr

• Transform the following equations from polar coordinates to rectangular coordinates.

3sin

4cos

r

r

Polar Equations and Graphs

• An equation whose variables are polar coordinates is called a polar equation.

• The graph of a polar equation consists of all points whose polar coordinates satisfy the equation.

#1

• Identify and graph the equation:

3r

#2

• Identify and graph the equation:

4

#3

• Identify and graph the equation:

sin 2r

Graphing a Polar Equation using your Calculator

1. Solve the equation for r in terms of .2. Clear the memory on your calculator.3. Select Mode - Polar. 4. Window ( step determines the number of

points that your calculator will graph)5. Zoom - Standard6. Enter the expression that you found in step 1.

#4

• Use your calculator to graph the polar equation:

sin 2r

#5

• Use your calculator to graph the polar equation:

cos 3r

• Let a be a nonzero real number. Then the graph of the equation is a horizontal line a units above the pole if a > 0 and units below the pole if a < 0 .

• The graph of the equation is a vertical line a units to the right of the pole if a > 0 and units to the left of the pole if a < 0.

sinr a a

cosr a

a

#6

• Identify and graph the equation:

4sinr

#7

• Identify and graph the equation:

2cosr

• Specific Types of Polar Graphs• Symmetry• Plot each point given in polar coordinates and

find other polar coordinates of the point.