 Modified 2020-04-13T10:02:43Z

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

r^2 = x^2 + y^2

And since $$r$$ equals 1, we can write it like

1 = x^2 + y^2
Figure 1: Unit circle creating sine and cosine waves. The unit circle is colored magenta, the radius $$r$$ yellow, the angle $$t$$ 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(π/2) = 1
\sin(3π/2) = -1
\cos(0) = \cos(2π) = 1
\cos(π) = -1
\cos(π/2) = \cos(3π/2) = 0

Since x^2 + y^2 = 1:

\cos(t)^2 + \sin(t)^2 = 1

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

\cos(t) = \sin(t + π/2)

# References 