Revised 2022-04-14T14:22:03Z

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.

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

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