What is the *unit circle*? What are the relations between the unit circle and trigonometric functions? What about Pythagoras?

# Definition

The unit circle is a circle with a radius \(r\) equal to 1. The unit circle will be centered at the origin in a coordinate system.

# Visualisation

In figure 1 below you can see an animation of the radius \(r\) rotating along the unit circle and creating an angle \(t\). The \(x\) value, which equals `\cos(t)` and the \(y\) value, which equals `\sin(t)`, will also be indicated.

The rotation with constant velocity will at the same time create:

- a sine wave from the current value of `y = \sin(t)`
- a cosine wave from the current value of `x = \cos(t)`

Thanks to Pythagoras, we have these relations between the variables

And since \(r\) equals 1, we can write it like

# Trigonometric functions

Since 360° equals 2π radians, we also have these beauties:

Since `x^2 + y^2 = 1`:

We can see that the sine wave lies exactly π/2 radians after the cosine wave, which imply that