# 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.

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. [See more in the Vectors in 2 Dimensions section.]

### Later, on this page...

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:

### Example 1

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

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

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

### Need Graph Paper?

### Example 2

Plot the points on the following polar grid:

a) (2, 60°)

b) (4, 165°)

c) (3, 315°)

Answer

## Converting Polar and Rectangular Coordinates

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

We've created an Polar to Rectangular Calculator that makes converting complex numbers in polar form to rectangular form easy if you don't have a handheld calculator.

We've written an overview for how to convert rectangular and polar forms using various handheld calculators in the complex numbers chapter.

From Pythagoras, we have: *r*^{2} = *x*^{2} + *y*^{2} and basic trigonometry gives us:

`tan\ theta=y/x` *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 3

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

Answer

Using calculator, we have:

`(2.35, -7.81)` rectangular ≡ `(8.16, -73.3^@)`

**Sketch** to check your answer!

(The sign " ≡ " means "**is identically equal to**".)

### Example 4

Convert the polar coordinates given by `(4.27, 168^@)` into rectangular coordinates.

Answer

Using calculator, we have:

(4.27, 168

^{o}) polar ≡ (−4.18, 0.888) rectangular

**Sketch** to check your answer!