7. Polar Coordinates

For certain functions, rectangular coordinates (those using x-axis and y-axis) are very inconvenient. In rectangular coordinates, we describe points as being a certain distance along the x-axis and a certain distance along the y-axis.

polar graph
A graph using
polar coordinates

But certain functions are very complicated if we use the rectangular coordinate system. Such functions may be much simpler in the polar coordinate system, which allows us to describe and graph certain functions in a very convenient way.

Polar coordinates work in much the same way that we have seen in trigonometry (radians and arc length, where we used r and θ) and in the polar form of complex numbers (where we also saw r and θ).

Vectors also use the same idea.

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Converting polar and rectangular coordinates

In polar coordinates, we describe points as being a certain distance (r) from the pole (the origin) and at a certain angle (θ) from the positive horizontal axis (called the polar axis).

The coordinates of a point in polar coordinates are written as

(r, θ)

The graph of the point (r, θ) is as follows:

math expression

 

Example

The point described in polar coordinates by (2, 3π/4) would look like this:

math expression

We use polar graph paper for drawing points in polar coordinates.


NOTE: Angles can be in degrees or radians for polar coordinates.


Exercise

Need Graph Paper?

rectangular grid
Download graph paper

(Polar graph paper included.)

Plot the points on the following polar grid:

a) (2, 60°) b) (4, 165°) c) (3, 315°)

polar grid


Answer


Converting Polar and Rectangular Coordinates

The conversion from polar to rectangular coordinates is the same idea as converting rectangular from to polar form in complex numbers.

[See how to convert rectangular and polar forms in the complex numbers chapter.]

math expression

From Pythagoras, we have: r2 = x2 + y2 and basic trigonometry gives us:

math expression x = r cos θ y = r sin θ

 

So it is the same type of thing that we had with complex numbers.

We can use calculator directly to find the equivalent values.

 

Example 1

Convert the rectangular coordinates given by (2.35, -7.81) into polar coordinates.


Answer


Example 2

Convert the polar coordinates given by (4.27, 168°) into rectangular coordinates.


Answer




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