Unit Circle

The simplest way to define the unit circle is to say it is the collection of all points that are one unit away from the origin.

Pretend is that an old fashioned clock is in your circle. Each number is a distance of unit one away from the center.

We can get a unit circle in a 2D Cartesian coordinate system by declaring

$x^2 + y^2 = 1$

With some mathematical wrangling we can change the above to

$y = +/- \sqrt {1 - x^2}$

Half the points of our unit circle fit the definition

$(x, \sqrt {1 - x^2})$

and the other half fit

$(x, - \sqrt {1 - x^2})$

For the above math to work correctly, it is necessary for ‘x’ to only take values between -1 and 1.

We can build a unit circle in 2D polar coordinates $(r,\theta)$ using all points that fit the following:

$(1,\theta)$

We can also plot points in 2D Cartesian Coordinates using the following two formulas: