Polar coordinates is a coordinate system that uses the complex numbers to describe a point in space. A point in space can be described with just a radius or just an angle, but **polar coordinates combines** the two into one coordinate.

The radius is called the amplitude and describes the length of the line extending from the origin (0, 0) to the point (x, y). The angle is called the phase and describes how far around the circle (0, 0) the point lies.

Changing the amplitude changes how long your line is and changes where the point lies on the plane. Changing the phase changes what axis of rotation your point lies on.

This tutorial will go into more detail about how to work with polar coordinates and how to convert between them and other *common coordinate systems*.

## The unit circle graph

In this graph, the x-axis and y-axis represent values on a coordinate plane. The x-* axis represents values* of the sin function and the y-axis represents values of the cos function.

All points on the circle have a value of 1 for one of the functions and -π/2 or π/2 for the other function.

The quadrants are divided by drawing a line from -π/2 to π/2. The quadrants are then named Sin-Pi, Cos-Pi, Sin-Tri, and Cos-Tri.

The points on the unit circle can be connected to *form many different shapes*. These shapes are important in understanding more *advanced math concepts like geometry* and algebra.

## How to find X and Y values

To find the values of X and Y, you **must first find** the sin and cos of the angles. The sine of an angle is a ratio between the length of the opposite side of the angle and the length of the adjacent side.

The cosine is a similar ratio between **two sides**, but it is typically between adjacent sides. Since we are dealing with quadrants, or quarters, of a circle, we will *use fractions* as our sin and cos numbers.

To find X, or what would be called x in math, you would take your sin value and put it over one divided by **two pi** (the number circumference is set to). To get Y, or what would be called y in math, you would take your cos value and put it over one divided by two pi as well.

## Examples

Several examples of trig functions in action are given below.

If a triangle has a 45 degree angle, then the sine of one half of the angle (1 2 Θ) is equal to the length of the side opposite the angle (x). This is because sin(1 2 Θ) = x, and since they are opposite sides, then x = sin(1 2 Θ) .

If a square has a 90 degree angle, then the sine of one half of the angle (1 2 Θ) is equal to the length of one side (x). This is because sin(1 2 Θ) = x, and since they are sides, then x = sin(1 2 Θ) .

If a circle has a pi (π) radius, then **cosine equals zero**. This is because cos(π / 2) = 0 , and since it is a radius, then 0 = cos(π / 2) .

## Practice problems

As mentioned before, **solving trig problems using inverse functions** can be very frustrating. Therefore, **practicing solving trig problems using inverse functions** is very important.

Problem solving practice works best when you have lots of practice problems to work with. Luckily, there are many websites and apps that have plenty of trig problem solving practice.

Most sites have you *solve basic trig problems first*, then move onto more advanced ones as you improve. Since most people find algebra harder than geometry, most sites start with basic algebra problems and then move onto more complex ones as you improve.

Some sites even have **real life problem solving cases** that you can work through to further strengthen your skills.