Definition of the unit circle

A unit circle is a circle whose radius is equal to 1. Furthermore, the circle has its center at the origin of a rectangular coordinate system.

Let P = (x , y) be a point on the circle. Then, make a right triangle by drawing a line perpendicular to x. The line is shown in green.

The horizontal leg of the triangle is x units away from the origin and the vertical leg of the triangle is y units away from the origin.

Using the pythagorean theorem, we get x² + y² = 1²    or    x² + y² = 1

We can take a step further. Let us name t the angle made with the radius in red and the x axis.

Take a close look at the triangle and you will see as we learned before that the adjacent side to angle t is x and the opposite side is y.

Therefore, sin(t) = y / 1 = y   and cos(t) = x / 1 = x

We can replace x = cos(t) and sin(t) in x² + y² = 1

We get cos(t)² + sin(t)² = 1

We just derived one of the most important trigonometric identities. We have just scratched the surface of what we can do with the unit circle. Next lesson will show that the unit circle can also be used to find sin (45 degrees).