Revised 2024-11-10T10:18:44Z

Unit circle

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

\[r^2 = x^2 + y^2\]

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

\[1 = x^2 + y^2\]

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, have more functionality and is easier to interact with.

Figure 1: Unit circle creating sine and cosine waves. (Use NNM Unit circle for an improved and standalone version.) The unit circle is colored magenta, the radius \(r\) yellow, the angle \(\theta\) (denoted \(t\) in figure) green, the \(x\) value red (point and line) and the \(y\) value blue (point and line). You can control the velocity (0-64 rpm) of the rotation, by clicking in the canvas and then using keys '+' and '-' respectively. You can toggle between displaying both waves, only sine wave and only cosine wave, by using key 'T'. You can toggle the display of the angle, by using key 'A'.

Trigonometric functions

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

\[\sin(0) = \sin(π) = \sin(2π) = 0\]

\[\sin(\frac{π}{2}) = 1\]

\[\sin(\frac{3π}{2}) = -1\]

\[\cos(0) = \cos(2π) = 1\]

\[\cos(π) = -1\]

\[\cos(\frac{π}{2}) = \cos(\frac{3π}{2}) = 0\]

Since \(x^2 + y^2 = 1\):

\[\cos^2 \theta + \sin^2 \theta = 1\]

We can see that the sine wave lies exactly \(\frac{π}{2}\) radians after the cosine wave, which imply that

\[\cos(\theta) = \sin(\theta + \frac{π}{2})\]

References

First published by Anders Gustafson 2018-03-17