Lesson POLAR COORDINATES

This lesson provide a brief overview of Polar Coordinates.

REFERENCES

http://www.random-science-tools.com/maths/coordinate-converter.htm
http://archives.math.utk.edu/visual.calculus/0/polar.6/index.html
http://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx
http://www.intmath.com/Plane-analytic-geometry/7_Polar-coordinates.php
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node5.html
http://www.engineeringtoolbox.com/converting-cartesian-polar-coordinates-d_1347.html
http://www.analyzemath.com/polarcoordinates/polar_rectangular.html
http://www.gap-system.org/~history/Extras/Coolidge_Polars.html
http://www.analyzemath.com/polarcoordinates/graphing_polar_equations.html
http://www.cliffsnotes.com/study_guide/Polar-Coordinates.topicArticleId-11658,articleId-11630.html


CARTESIAN COORDINATE SYSTEM

It is also called the rectangular coordinate system.

A point in this system is defined by its x-value and its y-value.

The point would be shown as a coordinate pair of (x,y).

The x-value is the number of units to the left or the right of the y-axis.

Each unit, or multiple of units, is represented by a vertical line

The y-value is the number of units above or below the x-axis.

Each unit, or multiple of units, is represented by a horizontal line.

An example of an equation in this coordinate system would be:

y =

An example of a point in this coordinate system would be:

(x,y) = (3,5)

A picture of that point in the Cartesian Coordinate Graph System is shown below:



POLAR COORDINATE SYSTEM

The graph of polar coordinates has a center called the Pole.

This Pole is surrounded by circles that represent the r-values on the graph.

This Pole is intersected with straight lines at various angles that represent T.

A horizontal line through the center represents 0 degrees on the right of the vertical line going through the pole, and 180 degrees on the left of the vertical line going through the pole.

A vertical line through the center represents 90 degrees above the horizontal line going through the pole, and 270 degrees below the horizontal line going through the pole.

It looks somewhat like the target of a dart board, if you are familiar with those.

A point in this system is defined by its r value and its T value.

The point would be shown as a coordinate pair of (r,T).

The r value is the number of units from the pole.

Each unit, or multiple of units, is represented by a circle surround the pole.

The T value is the number of degrees from the 0 degree line.

Each unit, or multiple of units, is represented by a line through the pole that is a specified number of degrees from the 0 degree line.

A picture of the Polar Coordinate System Graph is shown below:



An example of an equation in this coordinate system would be:

r = sin(T)

An example of a point in this coordinate system would be:

(r,T) = (5,60)

This point is 5 units from the center of the graph in the direction of a 60 degree angle measured counter-clockwise from the 0 degree mark.

A picture of that point in the Polar Coordinate Graph System is shown below:



GRAPH PAPER

If you do not have access to graph paper, you can print it from the following website.

http://www.printfreegraphpaper.com/

BASIC TRIGONOMETRIC FORMULAS USED IN CONVERTING FROM CARTESIAN COORDINATES TO POLAR COORDINATES AND VICE VERSA

The basic Trigonmetric Functions used are Sine, Cosine, Tangent.

If T is the angle, then:

Sin(T) = opposite / hypotenuse

Cos(T) = adjacent / hypotenuse

Tan(T) = opposite / adjacent.

In the Cartesian Coordinate System, the x-value represents the adjacent side and the y-value represents the opposite side.

These values are taken from the origin, which is the point (x,y) = 0,0).

The hypotenuse of the right triangle formed becomes the r value used in the Polar Coordinate System.

The angle formed between the lines created from the origin of the graph to the the x-value and the y-value becomes the T value in the Polar Coordinate System.

If you are in the Cartesian Coordinate System, then you use the following formulas to find the equivalent polar coordinate system point.

r = .

This is taken from the pythagorean formula of hypotenuse squared = adjacent squared plus opposite squared where r = hypotenuse, x = adjacent, y = opposite.

T = arctan(y/x).

This is taken from Tan(T) = opposite / adjacent where y = opposite and x = adjacent. Once you get the tangent, then you get the arctangent in order to find the angle whose tangent is y/x.

If you are in the Polar Coordinate System, then you use the following formulas to find the equivalent cartesian coordinate point.

x = r * cos(T)

This is taken from Cos(T) = adjacent / hypotenuse where x = adjacent and r = hypotenuse. Once you get cos(T) = x/r, then you multiply both sides of the equation by r to get r = x*cos(T).

y = r * sin (T)

This is taken from Sin(T) = opposite / hypotenuse where y = opposite and r = hypotenuse. Once you get sin(T) = y/r, then you multiply both sides of the equation by r to get r = y*sin(T).

A picture of what this looks like is shown below:



The derivation of X and Y in the graph is taken from the pythagorean formula.

x = = = x

The points used were from the lower left and lower right points on the graph.

y = = = y

The points used were from the upper right and lower right points on the graph.

CONVERTING A POINT IN CARTESIAN COORDINATES TO A POINT IN POLAR COORDINATES

Let the point in Cartesian Coordinates be (x,y) = (3,6)

We want to convert this to the same point expressed in Polar Coordinates.

First we want to find r.

r is the hypotenuse of the right triangle formed by x and y.

In the point (3,6), x = 3 and y = 6.

The length of x is equal to 3 as measured from the origin of the graph.

The length of y is equal to 6 as measured from the origin of the graph.

To get the length of r, we use the pythagorean formula to get which makes which makes = 9 + 36 which makes = 45 which makes r = which makes:

r = 6.708203933.

To find the angle T, we use the formula of tan(T) = y/x.

Since y = 6 and x = 3, we get tan(T) = 6/3 = 2.

Now that we have tan(T) = 2, we then take arctan(2) to find the angle whose tangent is equal to 2 to get:

arctan(2) = 63.43494882 degrees.

The Point whose Cartesian Coordinates are equal to (x,y) = (3,6) is equivalent to the point whose Polar Coordinates are (r,T) = (6.708203933, 63.43494882) where T is expressed in degrees.

GENERAL RULES FOR CONVERTING A POINT FROM CARTESIAN COORDINATES TO POLAR COORDINATES

Cartesian Coordinates are expressed as:

(x,y)

Polar Coordinates are expressed as:

(r,T)

To find r, you use the formula:

r =

To find T, you use the formula:

T = arctan (y/x)

CONVERTING A POINT IN POLAR COORDINATES TO A POINT IN CARTESIAN COORDINATES

Let the point in Polar Coordinates be (r,T) = 6.708203933, 63.43494882) where T is expressed in degrees.

We want to convert this to the same point expressed in Cartesian Coordinates

First we want to find x and y.

Since we know that Sin(T) = r/y, then we can find y by using the formula y = r*Sin(T).

Since we know that Cos(T) = r/x, then we can find x by using the formula x = r*Cos(T).

Plugging our values into this formula, we get:

y = 6.708203933 * Sin(63.43494882) = 6

x = 6.708203933 * Cos(63.43494882) = 3

The Point whose Polar Coordinates are equal to (r,T) = (6.708203933, 63.43494882), where T is expressed in degrees, is equivalent to the point whose Cartesian Coordinates are (x,y) = (3,6).

GENERAL RULES FOR CONVERTING A POINT FROM POLAR COORDINATES TO CARTESIAN COORDINATES

Polar Coordinates are expressed as:

(r,T)

Cartesian Coordinates are expressed as:

(x,y)

To find x, you use the formula:

x = r * Cos(T)

To find y, you use the formula:

y = r * Sin(T)

PICTURE OF EQUIVALENT POINTS IN CARTESIAN COORDINATES AND POLAR COORDINATES

The Cartesian Coordinates are (x,y) = (3,6).

The equivalent Polar Coordinates are (r,T) = (6.708203933, 63.43494882)

The picture of these points in each coordinate system is shown below:



CONVERTING THE ANGLE FROM DEGREES TO RADIANS

If T is expressed in degrees and you want to express it in radians, then you would divide T by 180 and multiply it by .

63.43494882 degrees would be equivalent to (63.43494882/180) * radians = .3562416382 * radians.

The point would then be shown as:

(r,T) = (6.708203933, .3562416382*) where T is expressed in radians.

is a constant that is equivalent to 3.141592654.

CONVERTING THE ANGLE FROM RADIANS TO DEGREES

If T is expressed in radians and you want to express it in degrees, then you would multiply T by 180 and divide it by .

.3562416382* radians * 180 / = .3562416382 * 180 = 63.43494882 degrees.

The in the numerator and the denominator cancels out.

CALCULATOR CONSIDERATIONS

Make sure your calculator is in the right mode.

If you are working with degrees, your calculator must be set to degree mode.

If you are working with radians, your calculator must be set to radian mode.

TOOL TO CHECK YOUR WORK IN CONVERTING FROM CARTESIAN TO POLAR AND IN CONVERTING FROM POLAR TO CARTESIAN

A tool to help you check your work in converting between cartesian and polar coordinates can be found at the following link.

http://www.random-science-tools.com/maths/coordinate-converter.htm

The link contains instructions on how to use the tool.

We will be using this tool in the next section.

OPTIONS FOR EXPRESSING A POINT IN POLAR COORDINATES

While a point in the Cartesian Coordinate System is expressed in only one way, there are several ways that a point in the Polar Coordinate System can be expressed.

You should know the different ways, even though you may not use them, because you will probably run into them at one point or another if you continue your studies in polar coordinate systems.

In the Cartesian Coordinate System, you express your points as (x,y). There are no alternate ways of expressing the same point.

In the Polar Coordinate System, you express your point as (r,T) = (5,60).

However, the same point (r,T) = (5,60) can also be expressed as:

(r,T) = (5, 60 +/- 360)

(r,T) = (-5, 60 +/- 180)

Each one of these references the same point.

Examples of (5, 60 +/- 360) would be:

(5,420), (5,-300)

Examples of (-5, 60 +/- 180) would be:

(-5,240), (-5,-120)

We will use some of these angles to show you how this works.

In our example, the point will be (r,T) = (5,60) in Polar Coordinates where T is expressed in degrees.

This point is equivalent to (x,y) = (2.5,4.330) in Cartesian Coordinates.

Any angle +/- 360 becomes an angle that points in the same direction as the original angle.

Any angle +/- 180 becomes an angle that points in the opposite direction that the original angle pointed in.

If you are not familiar with how this works, see my lesson on Trigonometric Functions of Angles greater than 90 degrees by clicking on the following link.

http://www.algebra.com/algebra/homework/playground/lessons/THEO-20091204.lesson

A picture of (r,T) = (5,60) is shown below:



You have already seen this picture earlier.

A picture of (r,T) = (5,-300) is shown below:



The angle of -300 degrees points in the same direction as the angle of 60 degrees.

Instead of going counter clockwise from the 0 degree mark to get to the 60 degree mark, we went clockwise 300 degrees to get to the same 60 degree mark.

A picture of (r,T) = (-5,240) is shown below:



When r is shown as negative, then the point is in the opposite direction indicated by the angle.

The angle is pointing down and to the left, but the point is 5 units from the center of the graph in the opposite direction (up and to the right).

To prove that all these polar coordinates that we just displayed reference the same point, we use the mechanized tool referenced earlier to convert from Polar Coordinates to Cartesian Coordinates.

That tool is referenced here again. It is at the following link:

http://www.random-science-tools.com/maths/coordinate-converter.htm

We input the following Polar Coordinates into the tool.

(r,T) = (5,60)
(r,T) = (5,-300)
(r,T) = (-5,240)

The following picture shows that the tool translated all of these Polar Coordinates to the same Cartesian Coordinate of:

(x,y) = (2.5,4.330)



GENERAL RULES FOR FINDING A POINT INDICATED BY A POLAR COORDINATE

If r is positive, then the point is in the same direction that the angle is pointing to.

If r is negative, then the point is in the opposite direction that the angle is pointing to.

GETTING THE ANGLE INTO THE PROPER RANGE SO YOU CAN SEE WHICH WAY IT'S POINTING ON THE POLAR COORDINATE GRAPH.

If the degree marks on the graph are from 0 to 360, then you need to translate your angle to get it in the range of the degree marks.

You do this by adding or subtracting 360 degrees from the angle until it gets in range.

Examples:

Assuming the range on the polar coordinate graph is 0 to 360 degrees, then:

60 degrees points in the direction of the 60 degree mark on the polar coordinate graph. No adjustment is necessary.

-300 degrees is out of range. Add 360 to it and you get 60 degrees which is in range. -300 degrees points in the direction of the 60 degree mark on the polar coordinate graph.

420 degrees is out of range. Subtract 360 from it and you get 60 degrees which is in range. 420 degrees points in the direction of the 60 degree mark on the polar coordinate graph.

240 degrees points in the direction of the 240 degree mark on the polar coordinate graph. No adjustment is necessary.

POLAR EQUATIONS

Equations in Polar form are equations that can be mapped on a Polar Coordinate Graph.

Converting an equation in Cartesian Form to an equation in Polar Form involves the use of equivalencies.

For example:

We know that x = r*cos(T) and we know that y = r*sin(T).

We can take the equation x^2 + y^2 = c and convert it into a Polar Equation as follows:

We replace x with r*cos(T) and we replace y with r*sin(T) to get:

(r*cos(T))^2 + (r*sin(T))^2 = c

We simplify to get:

r^2 * cos^2(T) + r^2 + sin^2(T) = c

We simplify further to get:

r^2 * (cos^2(T) + sin^2(T)) = c

Since we know that cos^2(T) + sin^2(T) = 1, then we get:

r^2 = c which becomes r = sqrt(c).

When we graph the equation of r = sqrt(c) on the Polar Coordinate System, we get the same circle as if we graphed the the equation of x^2 + y^2 = c on the Cartesian Coordinate System.

WHY USE A POLAR COORDINATE SYSTEM RATHER THAN A CARTESIAN COORDINATE SYSTEM

Some equations are much easier to represent in the Polar Coordinate System than they are in the Cartesian Coordinate System.

An example is the equation of a circle shown above.

In the Cartesian Coordinate System, the equation x^2 + y^2 = 25.

In order to graph this equation, we have to convert it into graphing form which would be y = sqrt(25-x^2).

In the Polar Coordinate System, the equation is r = 5.

Another example would be the graph of the following equation in Polar Coordinate Form. This equation would be much more complicated to represent in Cartesian Coordinate Form.



SOME COMMON EQUATIONS IN THE POLAR COORDINATE SYSTEM

Some of the references mentioned above contain pictures of the more commonly known equations in Polar Coordinate Form.

One such reference is shown below:

http://www.cliffsnotes.com/study_guide/Polar-Coordinates.topicArticleId-11658,articleId-11630.html

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