using polar coordinatespehs.psd202.org/documents/tfrey/1520535639.pdfconvert the rectangular...

Using Polar Coordinates

Graphing and

converting polar and

rectangular coordinates

Butterflies are among the most celebrated of all insects.

It’s hard not to notice their beautiful colors and graceful

flight. Their symmetry can be explored with trigonometric

functions and a system for plotting points called the polar

coordinate system. In many cases, polar coordinates are

simpler and easier to use than rectangular coordinates.

You are familiar with

plotting with a rectangular

coordinate system.

We are going to look at a

new coordinate system

called the polar

coordinate system.

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, )

radius angle

To plot the point

4,5

First find the angle

Then move out along

the terminal side 5

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

The polar coordinate system is formed by fixing a point, O,

which is the pole (or origin).

= directed angle Polar

axis

OPole (Origin)

The polar axis is the ray constructed from O.

Each point P in the plane can be assigned polar coordinates (r, ).

P = (r, )

r is the directed distance from O to P.

is the directed angle (counterclockwise) from the polar axis

to OP.

Graphing Polar Coordinates

The grid at the left is a polar grid. The

typical angles of 30o, 45o, 90o, … are

shown on the graph along with circles of

radius 1, 2, 3, 4, and 5 units.

Points in polar form are given as (r, )

where r is the radius to the point and is

the angle of the point.

On one of your polar graphs, plot the

point (3, 90o)?

A

The point on the graph labeled A is correct.

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

330

315

300

270240

225

210

180

150

135

120

0

9060

30

45

Polar coordinates can also be given with the angle in

degrees.

(8, 210°)

(6, -120°)

(-5, 300°)

(-3, 540°)


Now, try graphing .

Did you get point B?

Polar points have a new aspect. A radius

can be negative! A negative radius

means to go in the exact opposite

direction of the angle.

A

To graph (-4, 240o), find 240o and move 4

units in the opposite direction. The opposite

direction is always a 180o difference.

4

3,2

B

C

Point C is at (-4, 240o). This point could

also be labeled as (4, 60o).


How would you write point A with a

negative radius?

A correct answer would be (-3, 270o) or

(-3, -90o).

In fact, there are an infinite number of

ways to label a single polar point.

Is (3, 450o) the same point?

A

Don’t forget, you can also use radian angles

as well as angles in degrees.

B

C

On your own, find at least 4 different polar

coordinates for point B.


The point lies two units from the pole on the

terminal side of the angle

( , ) 2,3

r

.3

3

2,3

33,4

34

2

32

1 2 3

0

3 units from

the pole

Plotting Points


There are many ways to represent the point 2, .3

2

32

1 2 3

0

2,3

52,3

2 42,2 2,,3 3 3

52,3

additional ways

to represent the

point 2,3

( , ) , 2r r n

( , ) , (2 1)r r n

Let's plot the following points:

2,7

2,7

2

5,7

2

3,7

Notice unlike in the

rectangular

coordinate system,

there are many

ways to list the

same point.

Converting from Rectangular to Polar

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

To find the polar coordinate, we must

calculate the radius and angle to the

given point.(4, 3)

We can use our knowledge of right

triangle trigonometry to find the radius

and angle.

3

4

r

r2 = 32 + 42

r2 = 25

r = 5

tan = ¾

= tan-1(¾)

= 36.87o or 0.64 rad

The polar form of the rectangular point

(4, 3) is (5, 36.87o)


In general, the rectangular point (x, y) is converted to polar form (r, θ) by:

1. Finding the radius

(x, y)r2 = x2 + y2

y

x

r 2. Finding the angle

tan = y/x or = tan-1(y/x)

Recall that some angles require

the angle to be converted to the

appropriate quadrant.

However, the angle must be in the second

quadrant, so we add 180o to the answer

and get an angle of 123.70o.

The polar form is ( , 123.70o)

r2 = (-2)2 + 32

r2 = 4 + 9

r2 = 13

r =


o3156

2

3

2

3

1

.

tan

tan

13

On your own, find polar form for the point (-2, 3).

(-2, 3)

13

Converting from Polar to Rectanglar

322

34

304

x

x

rx

ocos

cos

2214

304

y

oy

ry

sin

sin

Convert the polar point (4, 30o) to rectangular coordinates.

4

30o

We are given the radius of 4 and angle of 30o.

Find the values of x and y.

Using trig to find the values of x and y, we know

that cos = x/r or x = r cos . Also, sin = y/r or

y = r sin .

x

y

The point in rectangular form is: 2,32

Converting from Polar to Rectanglar

2

3

2

13

3003

x

x

rx

ocos

cos

2

33

233

3003

y

oy

ry

sin

sin

On your own, convert (3, 5π/3) to rectangular coordinates.

-60o

We are given the radius of 3 and angle of 5π/3 or

300o. Find the values of x and y.

The point in rectangular form is:

2

33,

2

3


(r, )

(x, y)

Polex

y

(Origin)

y

r

x

The relationship between rectangular and polar

coordinates is as follows.

The point (x, y) lies on a

circle of radius r, therefore,

r2 = x2 + y2.

tanyx

cos xr

sinyr

Definitions of

trigonometric functions


Coordinate Conversion

cosx r cos xr

siny r sinyr

2 2 2r x y tanyx

(Pythagorean Identity)

Example:

Convert the point into rectangular coordinates. 4,3

1cos co3

24 s 42

x r

3sin sin 4 23 2

4 3y r

, 2, 2 3x y


Example:

Convert the point (1,1) into polar coordinates.

, 1,1x y

1tan 11

yx

4

2 2 2 21 1 2r x y

set of polar coordinates is ( , ) 2, .4

One r

5Another set is ( , ) 2, .4

r

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

3

r = 5

222 43 r

3

4tan

93.03

4tan 1

We'll find in radians

(5, 0.93)polar coordinates are:

Let's generalize this to find formulas for converting from

rectangular to polar coordinates.

(x, y)

r y

x

222 ryx

x

ytan

22 yxr

x

y1tan

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

coordinates.Based on the trig you

know can you see

how to find x and y?

44cos

x

rectangular coordinates are:

4,4

4y

x4

222

24

x

44sin

y

222

24

y 22,22

Let's generalize the conversion from polar to rectangular

coordinates.

r

xcos

,r

ry

x

r

ysin

cosrx

sinry

922 yx

Convert the rectangular coordinate system equation to a

polar coordinate system equation.

22 yxr 3r

r must be 3 but there is no

restriction on so consider

all values.

Here each r

unit is 1/2 and

we went out 3

and did all

angles.

? and torelated was

how s,conversion From

22 yxr

Before we do the conversion

let's look at the graph.

Convert the rectangular coordinate system equation to a

polar coordinate system equation.yx 42

cosrx

sinry

sin4cos2

rr

sin4cos22 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?

When trying to figure out the graphs of polar equations we

can convert them to rectangular equations particularly if

we recognize the graph in rectangular coordinates.

7r We could square both sides

492 rNow use our conversion:

222 yxr

4922 yxWe recognize this as a circle

with center at (0, 0) and a

radius of 7.

On polar graph paper it will centered at the origin and out 7

Let's try another:

3

Take the tangent of both sides

3tantan

Now use our conversion:

3x

y

We recognize this as a line with slope square root of 3.

3

x

ytan

Multiply both

sides by x

xy 3

To graph on a polar plot

we'd go to where

and make a line. 3

Let's try another: 5sin r

Now use our conversion:

We recognize this as a

horizontal line 5 units below

the origin (or on a polar plot

below the pole)

sinry

5y


Example:

Convert the polar equation into a rectangular

equation.

4sinr

4sinr

2 4 sinr r Multiply each side by r.

2 2 4x y y Substitute rectangular

coordinates.

22 2 4x y Equation of a circle with

center (0, 2) and radius of 2

Polar form

2 2 4 0x y y

Rectangular and Polar Equations

Equations in rectangular form use variables (x, y), while

equations in polar form use variables (r, ) where is an angle.

Converting from one form to another involves changing the

variables from one form to the other.

We have already used all of the conversions which are necessary.

Converting Polar to Rectangular

cos = x/r

sin = y/r

tan = y/x

r2 = x2 + y2

Converting Rectanglar to Polar

x = r cos

y = r sin

x2 + y2 = r2

Convert Rectangular Equationsto Polar Equations

The goal is to change all x’s and y’s to r’s and ’s.

When possible, solve for r.

Example 1: Convert x2 + y2 = 16 to polar form.

Since x2 + y2 = r2, substitute into the equation.

r2 = 16

Simplify.

r = 4

r = 4 is the equivalent polar equation to x2 + y2 = 16


Example 2: Convert y = 3 to polar form.

Since y = r sin , substitute into the equation.

r sin = 3

Solve for r when possible.

r = 3 / sin

r = 3 csc is the equivalent polar equation.


Example 3: Convert (x - 3)2 + (y + 3)2 = 18 to polar form.

Square each binomial.

x2 – 6x + 9 + y2 + 6y + 9 = 18

Since x2 + y2 = r2, re-write and simplify by combining like terms.

x2 + y2 – 6x + 6y = 0

Substitute r2 for x2 + y2, r cos for x and r sin for y.

r2 – 6rcos + 6rsin = 0

Factor r as a common factor.

r(r – 6cos + 6sin ) = 0

r = 0 or r – 6cos + 6sin = 0

Solve for r: r = 0 or r = 6cos – 6sin

Convert Polar Equationsto Rectangular Equations

The goal is to change all r’s and ’s to x’s and y’s.

Example 1: Convert r = 4 to rectangular form.

Since r2 = x2 + y2, square both sides to get r2.

r2 = 16

Substitute.

x2 + y2 = 16

x2 + y2 = 16 is the equivalent polar equation to r = 4


Example 2: Convert r = 5 cos to rectangular form.

Multiply both sides by r

r2 = 5x

Substitute for r2.

x2 + y2 = 5x is rectangular form.

r2 = 5 r cos

Substitute x for r cos


sin*

12r

Example 3: Convert r = 2 csc to rectangular form.

Since csc ß = 1/sin, substitute for csc .

Multiply both sides by 1/sin.

Simplify

y = 2 is rectangular form.

1

12

sin*

sin*sin r

You will notice that polar equations have graphs like the following:

Hit the MODE key.

Arrow down to where it says Func (short for "function" which is a bit misleading since they are all functions).

Now, use the right arrow to choose Pol.

Hit ENTER. (*It's easy to forget this step, but it's crucial: until you hit ENTER you have not actually selected Pol, even though it looks like you have!)

The calculator is now in polar coordinates mode. To see what that means, try this.

Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on.

In the r1= slot, type 5-5sin(θ)

Now hit the familiar X,T,θ,n key, and you get an unfamiliar result. In polar coordinates mode, this key gives you a θ instead of an X.

Finally, close off the parentheses and hit GRAPH.

If you did everything right, you just asked the calculator to graph the polar equation r=5-5sin(θ). The result looks a bit like a valentine.

The WINDOW options are a little different in this mode too. You can still specify X and Y ranges, which define the viewing screen. But you can also specify the θ values that the calculator begins and ends with.

Graph r = 3 sin 2θ

Enter the following window values:

Θmin = 0 Xmin = -6 Ymin = -4

θmax = 2π Xmax = 6 Ymax = 4

Θstep = π/24 Xscl = 1 Yscl = 1

Graph:

a. r = 2 cos θ

b. r = -2 cos θ

c. r = 1 – 2 cos θ


Each polar graph below is called a Limaçon.

1 2cosr 1 2sinr

–3

–5 5

3

–5 5

3

–3


Each polar graph below is called a Lemniscate.

2 22 sin 2r 2 23 cos2r

–5 5

3

–3

–5 5

3

–3


Each polar graph below is called a Rose curve.

2cos3r 3sin 4r

The graph will have n petals if n is odd, and 2n

petals if n is even.

–5 5

3

–3

–5 5

3

–3

a

a

Function Gallery in your book on page

352 summarizes all of the polar graphs.

You can graph these on your calculator. You'll need to

change to polar mode and also you must be in radians.

If you are in polar function mode when you hit your

button to enter a graph you should see r1 instead of y1.

Your variable button should now put in on TI-83's and

it should be a menu choice in 85's & 86's.

using polar coordinatespehs.psd202.org/documents/tfrey/1520535639.pdfconvert the rectangular...

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