Favicon NNM Elvira, March 2007 (Photo: Anders Gustafson) Favicon NNM
Modified 2021-06-06T14:01:52Z

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


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.


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:

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 a new improved implementation of the unit circle animation, namely the NNM Unit circle.

Figure 1: Unit circle creating sine and cosine waves. 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

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


First published by Anders Gustafson 2018-03-17
Logo for NNM
NNM Logo
Friends in need
CCF Cheetah Conservation Fund
NNM Clock
Random quote
Why worry about the Y10K problem if it is going to happen many centuries after your death? Exactly because you will already be dead, so the companies using your software will be stuck using your software without any other coder who knows the system well enough to come in and fix it.
Mozilla Developer Network contributors