Polar interactive p pt

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

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.

Rectangular Coordinates
For the point (-4, 6) we would move
4 units and 6 units

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.

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.

Polar Axis

Pole

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)

3 units
90º

Polar Coordinates
If we have the point (2, 135º) click where
you think the point will be.
Try Again, if your
struggling go back to the
description.

Great Job!

(2, 135º)

Polar Coordinates
If ϴ is negative, travel clockwise, like any
other negative angle.
Example: (2, -50º) (Click to here see where this will be.)

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

Polar Coordinates Sorry, Try Again.

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

Great Job!

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.

Converting Polar to
Rectangular
Think 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)
r
y
ϴ
x

Converting Polar to
Rectangular
Using trig we know:

and

(r, ϴ)
r
y
ϴ
x

Converting Polar to
Rectangular
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

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

First: Identify r and ϴ r = 4 and ϴ = 135º
Second: Find x by plugging r and ϴ x = r cos ϴ
into the cosine equation x = 4 cos135º
x = 4 (-√2 / 2)
x = -2√2

Find y by plugging r and y = r sin ϴ
Third: ϴ into the sine equation y = 4 sin 135º
y = 4 (√2 / 2)
y = 2√2

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

Converting Polar to
Rectangular
Try it on your own: Convert (-3, 30º) from
polar form to rectangular form. Click on the step
number to begin

First: r= -3 and ϴ= 30 Check step 1

-2.6
Approximate
Second: x=
all fractions to Check step 2
the nearest
tenth.
Third: y= Check step 3

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)
r
y
ϴ
x

Polar
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)
r
y
ϴ
x

Converting Rectangular to Polar

Example: Convert (5, -4) from polar form
to rectangular form. Click on the step number
to see how this works.

x = 5 and y = -4
First: Identify x and y
Second: Find r by plugging x and y
into the Pythagorean
equation

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

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

Converting Rectangular to Polar

Try it on your own: Convert (-3, 4) from
polar form to rectangular form. Click on the
step number to
begin.
First: x= and y= Check step 1

Second: r= Round answers to the
nearest tenth. Check step 2

Third: ϴ= Check step 3

Polar
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.

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:

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

r2 = r cos ϴ Time to replace what we can, with
those equations on the previous slide.
(click here to continue)
x2 + y2 = x If we complete the square we will get
the following equation in standard
(x - ½)2 + y2 = ¼ form. (click here to continue)

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

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

r = 4 sin ϴ
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

-3x+2
y=

Check your answer

Assessment
The following is a short ten-question quiz.

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

Question 1:
In polar coordinates, the origin is called the
__________ and the positive x-axis is
called the _____________.

Check your answers

Sorry, that is incorrect. See
Question 2: Polar Graphing for some help.

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

C.) D.)

Correct!

Question 3:
Which of the following points represents
(-3, 215º)? (Click on the appropriate letter.)
Sorry,
that is Correct!
incorrect. A
Look B
back at
graphing.

D
C

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

Sorry, that is incorrect.

Question 4: Look back at converting.

What rectangular coordinates represent
the polar coordinates (-2, 30º)?
This is still in
A.) (-1, -√3) B.) (-√3, -1) polar form.
Look back at
converting.

C.) (-√2, -√2) D.) (2, 210º)

Correct!


Question 5: Look back at converting.

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

A.) (√119, 67.3º) B.) (13, 67.3º)

C.) (√119, 85.2º) D.) (13, 85.2º)

Correct!

Question 6: Look back at polar
coordinates.

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

A.) (-5, 130º) B.) (-5, 490º)

C.) (-5, 310º) D.) (5, -130º)

Correct!

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

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!

Question Sorry, that is incorrect.
Look back at converting
8: equations.

Convert the following equation from polar
form to rectangular form. (Select the BEST answer.)
r = sin ϴ - cos ϴ

A.) 1 = y - x B.) (x + ½) 2 +(y - ½)2 = ½

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

Correct!

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

9:
Which graph represents the equation r = 4?
A.) B.)

Correct!
C.) D.)

Question Sorry, that is incorrect. Look
back at converting equations.
10:
Which graph represents the equation r cos ϴ = 4?

A.) B.)

Correct!
C.) D.)

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.)

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.

Polar interactive p pt

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