The Unit Circle

Introduction

The unit circle is a simple and unifying idea. The unit circle is defined as a circle of radius one centered at the origin of the Cartesian Coordinate System. The length of one is chosen, because it is simple and makes some of the math easier later on; All circle have the same properties.

Angle

A circle is a useful way to describe an angle as a fraction of the circumference. This, combined with the fact that the circumference of a circle is equal to 2πr, allows for every angle to be described by the radius of the circle. (see radians for more information)

Note: By convention angles are measured from the right-most edge of the circle, with the counter-clockwise direction as positive and the clockwise direction as negative.

Right Triangle

Every point along the circumference of the circle describes a corresponding right-triangle. The x and y component of the point correlates to the adjacent and opposite sides of the triangle. There are two things to notice here. First, every angle has a corresponding right triangle. And second, the sides of the triangle can both be positive and negative.

Notation

It is easy to be overwhelmed by the number of symbols used to describe the unit circle. The symbol θ (theta) is used to represent the angle. The two sides of the right-triangle correspond to the point (x,y). Finally, the radius is of length one.

Trigonometry

The first area of math that is illustrated by the unit circle is trigonometry. The trigonometric functions sine and cosine relate the angle of a right triangle to the ratio of its sides.

In the case of the unit circle, the hypotenuse will always be one, the adjacent side corresponds with the x coordinate, and the opposite side corresponds with the y coordinate.

Substituting the corresponding components simplifies both equations and creates a beautiful connection. See the interactive below:

Special Triangles

Typically, students are tasked with remembering a number of points on the unit circle. However, it is important to recognize that there are infinite points along the circumference of the circle and that points associated with the task are simply special cases. To build the chart below, see how to construct the unit circle chart.

Pythagorean Theorem

Another well known property of the right triangle is Pythagorean’s Theorem.

In the case of the unit circle: the adjacent side of the triangle, a, corresponds with x. The opposite side of the triangle, b, corresponds with y. And the hypotenuse, c, corresponds with the number one. Substituting the corresponding values into the Pythagorean equation results in the equation below.

Note, this is the equation of a circle of hypotenuse one in the Cartesian Coordinate system. Next, substitute in for x and y the trigonometric formulas found above:

The result is Pythagorean’s identity. A rather useful equation.

Onward

The unit circle does not end here. Depending on the reader, this fact may surface a large array of emotions. Part of mathematics is following patterns and logic, sometimes without any reason why or to what they apply to. Luckily, everything covered here is cornerstone to large areas of mathematics. There are many trigonometric formula left to be derived and a couple other things that can be related in surprising ways, like Euler’s number e and the imaginary number i… it’s a little crazy. For now, here are a couple of references and further readings.

References & Readings