Polar and Cartesian Coordinates

... and how to convert between them.

In a hurry? Read the Summary. But please read why first:

To pinpoint where we are on a map or graph there are two main systems:

Cartesian Coordinates

Using Cartesian Coordinates we mark a point by how far along and how far up it is:

coordinates cartesian (12,5)

Polar Coordinates

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

coordinates polar 13 at 22.6 degrees

Converting

To convert from one to the other we will use this triangle:

coordinates triangle

To Convert from Cartesian to Polar

When we know a point in Cartesian Coordinates (x,y) and we want it in Polar Coordinates (r,θ) we solve a right triangle with two known sides.

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

coordinates to polar

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

r2 = 122 + 52
r = √ (122 + 52)
r = √ (144 + 25)
r = √ (169) = 13

Use the Tangent Function to find the angle:

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

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

calculator-sin-cos-tan

What is tan-1 ?

It is the Inverse Tangent Function:

 

Summary: to convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

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 Cartesian

When we know a point in Polar Coordinates (r, θ), and we want it in Cartesian Coordinates (x,y) we solve a right triangle with a known long side and angle:

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

to cartesian coordinates

Use the Cosine Function for x:   cos( 22.6° ) = x / 13
Rearranging and solving:   x = 13 × cos( 22.6° )
    x = 13 × 0.923
    x = 12.002...
     
Use the Sine Function for y:   sin( 22.6° ) = y / 13
Rearranging and solving:   y = 13 × sin( 22.6° )
    y = 13 × 0.391
    y = 4.996...

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

 

How to Remember

(x,y) are alphabetical, and so are (cos,sin)

 

Summary: to convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :

 

But What About Negative Values of X and Y?

Quadrants

Four Quadrants

When 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 θ = 195°

So the point is at (−11.59, −3.11), which is in Quadrant III

But when converting from Cartesian to Polar coordinates ...

... the calculator can give the wrong value of tan-1

It all depends what Quadrant the point is in! Use this to fix things:

Quadrant Value of tan-1
I Use the calculator value
II Add 180° to the calculator value
III Add 180° to the calculator value
IV Add 360° to the calculator value

polar example 1

Example: P = (−3, 10)

P is in Quadrant II

The calculator value for tan-1(−3.33...) is −73.3°

The rule for Quadrant II is: Add 180° to the calculator value
θ = −73.3° + 180° = 106.7°

So the Polar Coordinates for the point (−3, 10) are (10.4, 106.7°)

polar example 2

Example: Q = (5, −8)

Q is in Quadrant IV

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

The rule for Quadrant IV is: Add 360° to the calculator value
θ = −58.0° + 360° = 302.0°

So the Polar Coordinates for the point (5, −8) are (9.4, 302.0°)

 

Summary

To convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :

To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

The value of tan-1( y/x ) may need to be adjusted:

 

Activity: A Walk in the Desert 2