Polar interactive p pt

POLAR V. RECTANGULAR COORDINATES AND EQUATIONS

Nicole Slosser

EDU 643

26 April 2011

Table of Contents

Rectangular Coordinates What are Polar Coordinates? Graphing Polar Coordinates Converting Polar to Rectangular Converting Rectangular to Polar Assessment Project

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Rectangular Coordinates

We are accustom to graphing with rectangular coordinates. When we see (3, -4) we know that we must go 3 units right and 4 units down using an xy-plane.

Click to here see the motion of the point.Next

Rectangular Coordinates

For the point (-4, 6) we would move

4 units and 6 units

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Check your answers

Polar Coordinates

Some things, such as navigation, engineering, and modeling real-world situations can’t easily be measured linearly.

We have polar coordinates to describe curves and rotations.

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Polar Coordinates

All polar coordinates begin with a pole (much like the origin). Click here to show the pole

And a polar axis (think positive x-axis, like where all trig angles start.) Click here to show the polar axis.

Pole

Polar Axis

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Polar Coordinates

Polar Coordinates are written (r, ϴ) where r is the distance from the pole (like a radius) and ϴ is the angle measure from the polar axis.

So the point (3, 90º) will look like this. (click here to see animation)

90º3 units

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Polar Coordinates

If we have the point (2, 135º) click where you think the point will be.

(2, 135º)

Try Again, if your struggling go back to the description.

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Great Job!

Polar Coordinates

If ϴ is negative, travel clockwise, like any other negative angle.

Example: (2, -50º) (Click to here see where this will be.)

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50º

Polar Coordinates

If r is negative, this means to move in the opposite direction. So you face the angle where you moved and will travel backwards from there.

Example: (-3, 45º) would look like this. (Click here to see motion.)

45º-3 units

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Polar Coordinates

Click where you would find the point (-3, 60º)

Great Job!

Sorry, Try Again.

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Converting Polar to Rectangular

Often it is useful to be able to go between the two graphing systems. We will use trig to help us convert from polar to rectangular.

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Converting Polar to RectangularThink of the point (r, ϴ) anywhere in the polar

plane.

We can create a right triangle that is x units horizontally and y units vertically. We can call the hypotenuse r because it is the distance from the origin/pole and the angle will be ϴ. (Click here to see the triangle.)

(r, ϴ)

yr

x Nextϴ


Using trig we know:

(r, ϴ)

yr

x

and

ϴ

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Solving both equations for x or y, we get:

x = r cos ϴ and y = r sin ϴ

We can use both of these equations to convert any point in polar form to rectangular form

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Example: Convert (4, 135º) from polar form to rectangular form.

First: Identify r and ϴ

Second:

r = 4 and ϴ = 135º

Find x by plugging r and ϴ into the cosine equation

x = r cos ϴx = 4 cos135ºx = 4 (-√2 / 2)x = -2√2

Third: Find y by plugging r and ϴ into the sine equation

y = r sin ϴy = 4 sin 135ºy = 4 (√2 / 2)y = 2√2

So, we have the point (-2√2, 2√2) in the xy-plane. Next


Try it on your own: Convert (-3, 30º) from polar form to rectangular form.

First:

Second:

Third:

r = and ϴ =

x =

y =

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Approximate all fractions to the nearest tenth.

Click on the step number to begin

Check step 1

Check step 2

Check step 3

-3 30

-2.6

Converting Rectangular to Polar

In order to convert the other way, from rectangular to polar, we have to use trig and that same right triangle.

(x, y)

yr

xϴ

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Suppose we have the point (x, y). Now we want to find r and ϴ in terms of x and y.

In order to find r, we have to find the length of the hypotenuse. (Quietly thank Pythagoras). We know

And right triangle trig tells us that

(x, y)

yr

xϴ

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Example: Convert (5, -4) from polar form to rectangular form.

First: Identify x and y

Second: Find r by plugging x and y into the Pythagorean equation

Third: Find ϴ by plugging x and y into the tangent equation

So, we have the point (√41, 321.3°) in the polar plane.

x = 5 and y = -4

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Click on the step number to see how this works.


Try it on your own: Convert (-3, 4) from polar form to rectangular form.

First:

Second:

Third:

x = and y =

r =

ϴ =

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Click on the step number to begin.

Round answers to the nearest tenth.

Check step 1

Check step 2

Check step 3


Whenever your point lies in Quadrant 2 or 3, you must add 180º to your new ϴ.

This makes the adjustment that your calculator doesn’t. Your calculator looks for the first ϴ, not necessarily the correct ϴ.

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Converting Equations

When converting equations from polar form into rectangular form (where we know how to graph it better) we look for the following two equations:

x = r cos ϴ and y = r sin ϴ

Also be on the lookout for:

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Example:

r = cos ϴ

Now this looks pretty close to what we want, but there is no r in front of cos ϴ. So let’s multiply both sides by r. (click here to continue)

r *( )

r2 = r cos ϴ Time to replace what we can, with those equations on the previous slide. (click here to continue)

x2 + y2 = xIf we complete the square we will get the following equation in standard form. (click here to continue)(x - ½)2 + y2 = ¼

So our equation is a circle with a center at (½, 0) and a radius of ½.

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Your turn: Convert the following equation in polar form into rectangular form.

r = 4 sin ϴ

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2 2 + =

Check your answers


Try that one more time: Convert the following equation from polar form to rectangular form

4r sin ϴ + 12r cos ϴ = 8

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y =

Check your answer

-3x+2

Assessment

The following is a short ten-question quiz.

Please click on the letter that best matches the correct answer.

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Question 1:

In polar coordinates, the origin is called the __________ and the positive x-axis is called the _____________.

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Check your answers

Question 2:

Which graph below represents the polar coordinate (-2, 330º)?

A.) B.)

C.) D.)

Correct!

Sorry, that is incorrect. See Polar Graphing for some help.

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Question 3:

Which of the following points represents (-3, 215º)? (Click on the appropriate letter.)

A

B

CD

Correct!Sorry, that is incorrect. Look back at graphing.

Notice that r is negative. Look back at what this means here. Next

Question 4:

What rectangular coordinates represent the polar coordinates (-2, 30º)?

A.) (-1, -√3) B.) (-√3, -1)

C.) (-√2, -√2) D.) (2, 210º)

Correct!

Sorry, that is incorrect. Look back at converting.

This is still in polar form. Look back at converting.

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Question 5:

What is the polar form of the rectangular coordinate (12, 5)?

A.) (√119, 67.3º) B.) (13, 67.3º)

D.) (13, 85.2º) C.) (√119, 85.2º)

Correct!


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Question 6:

What is another polar coordinate that also corresponds to (5, 130º)?

A.) (-5, 130º) B.) (-5, 490º)

D.) (5, -130º)C.) (-5, 310º)

Correct!

Sorry, that is incorrect. Look back at polar coordinates.

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Question 7:Find the polar coordinates for the rectangular

point (-1.3, -2.1). (Round your answer to the nearest hundredth.)

A.) (2.47, 58.24º) B.) (2.47, 238.2º)

C.) (-2.47, 238.2º) D.) (2.47, -58.24º)

Correct!


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Question 8:Convert the following equation from polar

form to rectangular form. (Select the BEST answer.)

r = sin ϴ - cos ϴ

A.) 1 = y - x

D.) (x + ¼) 2 +(y - ¼)2 = ½C.) x 2 + y2 = y - x

B.) (x + ½) 2 +(y - ½)2 = ½

Correct!

Sorry, that is incorrect. Look back at converting equations.

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Question 9:Which graph represents the equation r = 4?

A.) B.)

C.) D.)

Correct!


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Question 10:Which graph represents the equation r cos ϴ = 4?

A.) B.)

C.) D.)

Correct!


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Project

Rummaging through a friend’s attic, a treasure map was discovered. Lucky day! But at a closer inspection you realize that the map is for Anchorage, Alaska and it is more of a list of directions than a map. It appears that all of the directions are in polar coordinates. But, Anchorage isn’t laid out that way.

In 1964, Anchorage was struck by a large earthquake which damaged at great deal of the town. The local government decided it would be best to rebuild from scratch. Roads were designed so that they always crossed at right angles to improve traffic. Anchorage is built like a huge rectangular grid.

It appears that the directions begin from your friend’s old family home, which used to be located on the corner of present-day 7th Street and G Street, one block south of the current the Alaska Center of Performing Arts. Your mission is to find the location of the treasure. (Locate the cross-streets on the modern map.) (Search for the corner of 7th and G, Anchorage, AK.)

Here are the map directions: Travel (3, 45°) from there, travel (5.83, 210.96°) from there, travel (-3.16, 108.43°) from there (4.47, 386.57°) and finally travel (6.40, -231.34°).

And your follow-up assignment: Paris was designed long before cars and updated traffic constraints. The city is laid out much like a polar grid. With a partner, design a treasure map with a difficult-to-locate solution to challenge another pair. (Include at least four steps and a correct solution on a separate sheet of paper.) Next

http://maps.google.com/

Sources:“Corner of 7th and G Anchorage, AK.” Google Maps. 6 April 2011

<http://maps.google.com/>.

Gurewich, Nathan and Ori Gurewich. Teach Yourself Visual Basic 4 in 21 days: Third Edition. Sams Publishing. Indianapolis, IN. 1995.

“History of Anchorage, Alaska.” Wikipedia, the free encyclopedia. 5 April 2011 <http://en.wikipedia.org/wiki/History_of_Anchorage,

_Alaska>.

Sullivan, Michael. Precalculus: Eighth Edition. Pearson: Prentice Hall. Upper Saddle River, NJ. 2008.

All graphics drawn and animated by Nicole Slosser using PowerPoint tools.

Education

Polar interactive p pt