The unit circle is a circle of radius 1 that we can use to pull together Pythagoras and trigonometry. The unit circle has the equation:
(1) 
This looks like this:
If we project a line from the origin to a point 
on the unit circle we get:
If we construct a triangle from this line to the x and y axis, we can use the trigonometric functions to find the values for the coordinates of the point on the unit circle. Remembering that the adjacent is related to the hypotenuse via:
(2) 
With the hypotenuse having a value of 1 we can see that:
(3) 
So that gives us 
. We can do the same for the opposite, using the sine of 
(4) 
Again the hypotenuse is 1, so we have:
(5) 
Which gives us 
.
We can now combine those results with Pythagoras’s theorem to see that any line from the origin at 
to a point on the unit circle gives:
(6) 
So on a diagram we have the following:
From the diagram we can see that the triangle formed when we project a line from the origin to a point on the circle will give side lengths related to the trigonometric ratios. The base or adjacent is 
and the side or opposite is 
. So the unit circle ties together Pythagoras and the trigonometric ratios for 
and 
. Any point on the unit circle has the coordinates 
.
