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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

Unit Circle from http://commons.wikimedia.org/wiki/File:Unit_circle_angles.svg

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.
ruler

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.
Unit Circle Conversion ruler

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.

Unit Circle as a Pizza

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.

Unit Circle in Sections

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.

Unit Circle in Four Sections

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.