Polar and Cartesian Coordinates... and how to convert between them. To pinpoint where you are on a map or graph there are two main systems:

Cartesian CoordinatesUsing Cartesian Coordinates you mark a point by how far along and how far up it is:

Polar CoordinatesUsing Polar Coordinates you mark a point by how far away, and what angle it is:

Converting

To convert from one to the other, you need to solve the triangle:

To Convert from Cartesian to PolarWhen you know a point in Cartesian Coordinates (x,y) and want it in Polar Coordinates (r,) you solve a right triangle with two known sides.

Example: What is (12,5) in Polar Coordinates?

Use Pythagoras Theorem to find the long side (the hypotenuse):

r2 = 122 + 52 r = (122 + 52) r = (144 + 25) = (169) = 13Use the Tangent Function to find the angle:

tan( ) = 5 / 12 = tan-1 ( 5 / 12 ) = 22.6 (to one decimal)

What is tan-1 ? It is the Inverse Tangent Function. Tangent takes an angle and gives you a ratio, Inverse Tangent takes a ratio (like "5/12") and gives you an angle.

Answer: the point (12,5) is (13, 22.6) in Polar Coordinates.

So to convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,): r =(x +y ) = tan-1 2 2

(y/x)

Note: Calculators may give the wrong value of tan-1 () when x or y are negative ... see below for more.

To Convert from Polar to CartesianWhen you know a point in Polar Coordinates (r, ), and want it in Cartesian Coordinates (x,y) you solve a right triangle with a known long side and angle:

Example: What is (13, 22.6) in Cartesian Coordinates?

Use the Cosine Function for x: Rearranging and solving:

cos( 22.6 ) = x / 13 x = 13 cos( 22.6 ) = 13 0.923 = 12.002...

Use the Sine Function for y: Rearranging and solving:

sin( 22.6 ) = y / 13 y = 13 sin( 22.6 ) = 13 0.391 = 4.996...

Answer: the point (13, 22.6) is almost exactly (12, 5) in Cartesian Coordinates.

So, to convert from Polar Coordinates (r,) to Cartesian Coordinates (x,y) : x = r cos( ) y = r sin( )

But What About Negative Values of X and Y? Four QuadrantsWhen we include negative values, the x and y axes divide the space up into 4 pieces: Quadrants I, II, III and IV (They are numbered in a counter-clockwise direction)

When converting from

Polar to Cartesian coordinates it all works out nicely:

Example: What is (12, 195) in Cartesian coordinates?r = 12 and = 195 x = 12 cos(195) = 12 -0.9659... = -11.59 to 2 decimal places y = 12 sin(195) = 12 -0.2588... = -3.11 to 2 decimal places So the point is at (-11.59, -3.11), which is in Quadrant III But when converting from

Cartesian to Polar coordinates,

... the calculator may give you the wrong value of tan-1

It all depends what Quadrant the point is in! Use this to fix things: Quadrant I II III IV Value of tan-1 Use the calculator value Add 180 to the calculator value Add 180 to the calculator value Add 360 to the calculator value

Example: P = (-3, 10)P is in Quadrant II r = ((-3)2 + 102) = 109 = 10.4 to 1 decimal place = tan-1(10/-3) = tan-1(-3.33...) The calculator value for tan-1(-3.33...) is -73.3

The rule for Quadrant II is: Add 180 to the calculator value = -73.3 + 180 = 106.7So the Polar Coordinates for the point (-3, 10) are (10.4, 106.7)

Example: Q = (5, -8)Q is in Quadrant IV r = (52 + (-8)2) = 89 = 9.4 to 1 decimal place = tan-1(-8/5) = tan-1(-1.6)

The calculator value for tan-1(-1.6) is -58.0

The rule for Quadrant IV is: Add 360 to the calculator value = -58.0 + 360 = 302.0So the Polar Coordinates for the point (5, -8) are (9.4, 302.0)

SummaryTo convert from Polar Coordinates (r,) to Cartesian Coordinates (x,y) : x = r cos( ) y = r sin( )

To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,): r = ( x 2 + y2 ) = tan-1 ( y / x )

The value of tan-1( y/x ) may need to be adjusted: Quadrant Quadrant Quadrant Quadrant I: Use the calculator value II: Add 180 III: Add 180 IV: Add 360

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Cartesian CoordinatesCartesian coordinates can be used to pinpoint where you are on a map or graph.

Cartesian CoordinatesUsing Cartesian Coordinates you mark a point on a graph by how far along and how far up it is:

The point (12,5) is 12 units along, and 5 units up.

X and Y AxisThe left-right (horizontal) direction is commonly called X.

The up-down (vertical) direction is commonly called Y.

Put them together on a graph ... ... and you are ready to go!

Where they cross over is the "0" point, you measure everything from there. The X Axis runs horizontally through zero The Y Axis runs vertically through zero Axis: The reference line from which distances are measured. The plural of Axis is Axes, and is pronounced ax-eez

And you can remember which is which by:

x is A CROSS, so x is ACROSS the page.Example:

Point (6,4) is 6 units across (in the x direction), and 4 units up (in the y direction) So (6,4) means:

Go along 6 and then go up 4 then "plot the dot".

Play With It !Now would be a good time to play with Interactive Cartesian Coordinates to see for yourself how it all works.

Like 2 Number Lines Put TogetherIt is like we put two Number Lines together, one going left-right, and the other going updown.

DirectionAs x increases, the point moves further right. When x decreases, the point moves further to the left.

As y increases, the point moves further up. When y decreases, the point moves further down.

Writing CoordinatesThe coordinates are always written in a certain order: the horizontal distance first, then the vertical distance.

This is called an "ordered pair" (a pair of numbers in a special order) And usually the numbers are separated by a comma, and parentheses are put around the whole thing like this:

(3,2)Example: (3,2) means 3 units to the right, and 2 units up Example: (0,5) means 0 units to the right, and 5 units up. In other words, only 5 units up.

The OriginThe point (0,0) is given the special name "The Origin", and is sometimes given the letter "O".

Abscissa and OrdinateYou may hear the words "Abscissa" and "Ordinate" ... they are just the

x and y values:

Abscissa: the horizontal ("x") value in a pair of coordinates: how far along the point is Ordinate: the vertical ("y") value in a pair of coordinates: how far up or down the point is

"Cartesian" ... ?They are called Cartesian because the idea was developed by the mathematician and philosopher Rene Descartes who was also known as Cartesius. He is also famous for saying "I think, therefore I am".

What About Negative Values of X and Y?Just like with the Number Line, you can also have negative values.

Negative: start at zero and head in the opposite direction: Negative x goes to the left Negative y goes down

So, for a negative number:

go backwards for x go down for y

For example (-6,4) means: go back along the x axis 6 then go up 4.

And (-6,-4) means:

go back along the x axis 6 then go down 4.

Four QuadrantsWhen we include negative values, the x and y axes divide the space up into 4 pieces: Quadrants I, II, III and IV (They are numbered in a counter-clockwise direction)

In Quadrant I both x and y are positive, but ... Like this: in Quadrant II x is negative (y is still positive), in Quadrant III both x and y are negative, and in Quadrant IV x is positive again, while y is negative.

QuadrantI II III IV

X Y Example (horizontal) (vertical)Positive Negative Negative Positive Positive Positive Negative Negative (-2,-1) (3,2)

Example: The point "A" (3,2) is 3 units along, and 2 units up.

Both x and y are positive, so that point is in "Quadrant I" Example: The point "C" (-2,-1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction). Both x and y are negative, so that point is in "Quadrant III"

Note: The word Quadrant comes form quad meaning four. For example, four babies born at one birth are called quadruplets, a four-legged animal is a quadruped. and aquadrilateral is a four-sided polygon.

Dimensions: 1, 2, 3 and more ...Think about this:

The Number Line can only go:

1

left-right

so any position needs just one number

Cartesian coordinates can go:

2

left-right, and up-down

so any position needs two numbers

How do we locate a spot in the real world (such as the tip of your nose)? We need to know:

3

left-right, up-down, and forward-backward,

that is three numbers, or 3 dimensions!

3 Dimensions

Cartesian coordinates can be used for locating points in 3 dimensions as in this example:

Here the point (2, 4, 5) is shown in three-dimensional Cartesian coordinates.In fact, this idea can be continued into four dimensions and more - I just can't work out how to illustrate that for you!

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