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 \(\theta\). The \(x\) value, which equals \(\cos(\theta)\) and the \(y\) value, which equals \(\sin(\theta)\), will also be indicated.

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

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

Thanks to Pythagoras, we have these relations between the variables

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

**NB! There is an improved implementation of the interactive unit circle animation, namely the NNM Unit circle, which I recommend to use instead of the animation below**. Both are working perfectly, but the *NNM Unit circle* is probably a bit more pedagogic.

# 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 \(\frac{π}{2}\) radians after the cosine wave, which imply that