How to Memorize Coordinates on the Unit Circle

Written by tutor Steve C.

Here is the Unit Circle, with the common radians and x & y coordinates listed

Image source here, used
with permission.

This can be found in almost every Algebra II,
Trigonometry, and Pre-Calculus
text book. It’s a great, except that – it can be a pain to learn, much less to memorize.

To solve this issue, I turn to one of my Dad’s favorite expressions: Divide and Conquer.
To make it easy to follow, let’s first look at one inch on a ruler, as it appears in most books.
Next to it, we see the same one inch, only this time, the values are grouped by denominator.

Notice how the values are the same on each table, but on the right, the values are listed along with all equal values with the same denominator. The left table only shows the fraction with the lowest denominator, but the values are all equal. By the time you study trigonometry, this idea will be understandable.

Now, when we apply the same principle to the unit circle, it becomes much easier to learn.

Notice how the values are the same on each ruler, but on the right, the values are listed along with all equal values with the same denominator. The left table only shows the fraction with the lowest denominator, but the values are all equal. Just like an inch ruler, this same concept works in trigonometry.

The unit circle can be seen as a pizza that can be cut a variety of ways.

Keeping in mind that the circumference of a circle is 2π, then the number of slices will determine the size of each slice of π (sorry).

The first circle is cut into 12 slices, with each being 2π/12 or π/6 of the pizza. Each piece is identified as a portion of π/6, along with
equal value with the lowest denominator.

The second circle is cut into 6 slices, with each being 2π/6 or π/3 of the pizza. Each piece is shown as a portion of π/3.

The third circle is cut into 4 slices, with each being 2π/4 or π/2 of the pizza. Each piece is shown as a portion of π/2.

By visually subdividing the circle as you would a pizza, the lowest denominator concept works in trigonometry just as it did with a simple 1 inch ruler.