Let’s say we have a point with coordinates on the unit circle, like the one shown here:

Using the Pythagorean Theorem, we know that . We also know this is true using the equation for a circle with radius 1 unit, , which is true for the point since it is on the circle.

Another way to write the coordinates of is to use the angle , which gives the location of on the unit circle relative to the point . Thinking of this way, its coordinates are . Since and , we can return to the Pythagorean Theorem and say that  is also true.

What if were a different angle and wasn’t in Quadrant I? It turns out that no matter the quadrant, the coordinates of any point on the unit circle given by an angle are . In fact, the definitions of and are the - and -coordinates of the point on the unit circle  radians counterclockwise from . Up until today, we’ve only been using the Quadrant I values for cosine and sine to find side lengths of right triangles, which are always positive.

This revised definition of cosine and sine means that is true for all values of defined on the unit circle and is known as the Pythagorean Identity.