11.1 Polar Coordinates and Graphs

Objective 1) To graph polar equations.2) To convert polar to rectangular3) To convert rectangular to polar

One way to give someone directions is to tell them to go three blocks East and five blocks South. This is like x-y Cartesian graphing.

Another way to give directions is to point and say “Go a half mile in that direction.”

Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle.

Initial ray

r A polar coordinate pair

determines the location of a point.

,r O

The center of the graph is called the pole.

Angles are measured from the positive x-axis.

Points are represented by a radius and an angle

(r, )To plot the point

4,5

First find the angle

Then move out along the terminal side 5

A negative angle would be measured clockwise like usual.

To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side.

4

3,3

3

2,4

2,7

2,7

2

5,7

2

3,7

Therefore unlike in the rectangular coordinate system, there are many ways to express the same point.

Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system.

(3, 4)

r

Based on the trig you know can you see how to find r and ?

4

3r = 5

222 43 r

3

4tan

93.03

4tan 1

We'll find in radians

(5, 0.93)polar coordinates are:

Convert Cartesian Coordinates to Polar Coordinates

Let's generalize this to find formulas for converting from rectangular to polar coordinates.

(x, y)

r

y

x

222 ryx

x

ytan

2 2r x y

x

y1tan

x = r cos, y = r sin

(b) (–1, 1) lies in quadrant II.

Since one possible value for is 135º. Also,

Therefore, two possible pairs of polar coordinates are

,1tan 11

.21)1( 2222 yxr

).225,2( and )135,2(

Giving Alternative Forms for Coordinates of a Point

Now let's go the other way, from polar to rectangular coordinates.

4cos ,4

x

rectangular coordinates are:

4,4

4 yx4

222

24

x

4sin ,4

y

222

24

y

Convert Polar Coordinates to Cartesian Coordinates

Let's generalize the conversion from polar to rectangular coordinates.

r

xcos

,r

r yx

r

ysin

cosrx

sinry

Convert Polar Coordinates to Cartesian Coordinates

Graphs of Polar Equations

• Equations such as

r = 3 sin , r = 2 + cos , or r = ,

are examples of polar equations where r and are the variables.

• The simplest equation for many types of curves turns out to be a polar equation.

• Evaluate r in terms of until a pattern appears.

Find a rectangular equation for r = 4 cos θ

yx 42

cosrx

sinry sin4cos 2 rr

sin4cos 22 rr

substitute in for x and y

We wouldn't recognize what this equation looked like in polar coordinates but looking at the rectangular equation we'd know it was a parabola.

What are the polar conversions we found for x and y?

Converting a Cartesian Equation to a Polar Equation

2

4sin

cosr

4 tan secr

(b) Solve the rectangular equation for y to get

(c)

.223

xy

Convert a Cartesian Equation to a Polar Equation

3x + 2y = 4Let x = r cos and y = r sin to get

.sin2cos3

4or4sin2cos3

rrr

Cartesian Equation Polar Equation

Convert r = 5 cos to rectangular equation.

Since cos = x/r, substitute for cos. 5x

rr

Multiply both sides by r, we haver2 = 5x

Substitute for r2 by x2 + y2, then

This represents a circle centered at (5/2, 0) and of radius 5/2 in the Cartesian system.

x² + y² = 5x

2r

ry

Now you try: Convert r = 2 csc to rectangular form.

Since csc = r/y, substitute for csc.

Multiply both sides by y/r.

Simplify, we have (a horizontal line) is the rectangular form. y = 2

2y r y

rr y r

For the polar equation

(a) convert to a rectangular equation,

(b) use a graphing calculator to graph the polar equation for 0 2, and

(c) use a graphing calculator to graph the rectangular equation.

(a) Multiply both sides by the denominator.

,sin14

r

4 sin 4

1 sinr r r

sin 4 4r r r y 2 24 (4 )r y r y

2 2 2(4 )x y y

Convert to a rectangular equation:

Multiply both sides by the denominator.

,sin14

r

4 sin 4

1 sinr r r

sin 4 4r r r y

2 24 (4 )r y r y

2 2 2(4 )x y y

2 2 2

2

2

16 88 168( 2)

x y y yx yx y

Square both sides.

rectangular equation

It is a parabola vertex at (0, 2) opening down and p = –2, focusing at (0, 0), and with diretrix at y = 4.

(b) The figure shows (c) Solving x2 = –8(y – 2)a graph with polar for y, we obtaincoordinates.

.2 281 xy

Theorem Tests for Symmetry

Symmetry with Respect to the Polar Axis (x-axis):

Theorem Tests for Symmetry

Theorem Tests for Symmetry

Symmetry with Respect to the Pole (Origin):

The tests for symmetry just presented are sufficient conditions for symmetry, but not necessary.

In class, an instructor might say a student will pass provided he/she has perfect attendance. Thus, perfect attendance is sufficient for passing, but not necessary.

Identify points on the graph:

Polar axis:

Symmetric with respect to the polar axis.

Check Symmetry of:

The test fails so the graph may or may not be symmetric with respect to the above line.

The pole:

The test fails, so the graph may or may not be symmetric with respect to the pole.

Cardioids (a heart-shaped curves)

are given by an equation of the form

r a(1 cos) r a(1 sin )

r a(1 cos) r a(1 sin )

where a > 0. The graph of cardioid passes through the pole.

Graphing a Polar Equation (Cardioid)

Example 3 Graph r = 1 + cos .

Analytic Solution Find some ordered pairs until a pattern is found.

r = 1 + cos r = 1 + cos

0º 2 135º .3

30º 1.9 150º .1

45º 1.7 180º 0

60º 1.5 270º 1

90º 1 315º 1.7

120º .5 360º 2

The curve has been graphed on a polar grid.

Limacons without the inner loop

are given by equations of the form

where a > 0, b > 0, and a > b. The graph of limacon without an inner loop does not pass through the pole.

0 5123

6

73.4

2

323

Let's let each unit be 1.

3

4

2

123

2

3023

3

2 22

123

6

527.1

2

323

1123 Since r is an even function of , let's plot the symmetric points.

This type of graph is called a limacon without an inner loop.

cos23r

Graph r = 3 + 2cos

Limacons with an inner loop

are given by equations of the form

where a > 0, b > 0, and a < b. The graph of limacon with an inner loop will pass through the pole twice. Ex: r = 1 – 2cosθ

Lemniscates

are given by equations of the form

and have graphs that are propeller shaped.Ex: r = 23 sin 2

Graphing a Polar Equation (Lemniscate)

Graph r2 = cos 2.

Solution Complete a table of ordered pairs.

0º ±1

30º ±.7

45º 0

135º 0

150º ±.7

180º ±1

2cosr

Values of for 45º < < 135º are not included because corresponding values of cos 2 are negative and do not have real square roots.

Rose curves

are given by equations of the form

and have graphs that are rose shaped. If n is even and not equal to zero, the rose has 2n petals; if n is odd not equal to +1, the rose has n petals. Ex: r = 2sin(3θ) and

r = 2sin(4θ)

Assignment

P. 400 #1 – 11 odd ( a and b is enough but can do all if want more practice)

330

315300

270240

225210

180

150

135120

0

9060

3045

Polar coordinates can also be given with the angle in degrees.

(8, 210°)

(6, -120°)

(-5, 300°)

(-3, 540°)

Give three other pairs of polar coordinates for the point P(3, 140º).

(3, –220º)

(–3, 320º)

(–3, –40º)

Since r is –4, Q is 4 units in the negative direction from the pole on an extension of the ray.

The rectangular coordinates:

32

221

4

32

cos4

x

3223

4

32

sin4

y

).32,2(

Plot each point by hand in the polar coordinate system. Then determine the rectangular coordinates of each point. 2

34,Q

Graphing a polar Equation Using a Graphing Utility

• Solve the equation for r in terms of θ.

• Select a viewing window in POLar mode. The polar mode requires setting max and min and step values for θ. Use a square window.

• Enter the expression from Step1.

• Graph.