Unit Circle Applications

The unit circle is a circle of radius one placed at the center of the coordinate system. There are many applications of the unit circle in trigonometry, physics and higher level math. This page summarizes some of the many applications of the unit circle.

Unit Circle (Trigonometry)

The unit circle visualizes the input and output of the trigonometric functions. As a concept, the unit circle expands the right triangle definitions of the trigonometric functions to all real numbers using the improved circle definitions of the functions as discussed on this page.

The unit circle demonstrates the output of the trigonometric functions sine, cosine and tangent.
Figure 1: Unit Circle Trigonometry

The property each trigonometric function corresponds to on the unit circle is shown in the figure above and summarized in the table below. These properties are useful when, for example, converting between polar and cartesian coordinates. They are used throughout the rest of this page.

Geometry Function
Geometry of the Sine Function Sine Function
The output of sine corresponds to the distance from a point on the perimeter of the unit circle to the -axis.
Geometry of the Cosine Function Cosine Function
The output of cosine corresponds to the distance from a point on the perimeter of the unit circle to the -axis.
Geometry of the Tangent Function Tangent Function
The output of tangent corresponds to the length of the line tangent to a point on the unit circle starting from the point and intersecting with the x-axis.
Geometry of the Cotangent Function Cotangent Function
The output of cotangent corresponds to the length of the line tangent to a point on the unit circle starting from the point and intersecting with the y-axis.

Unit Circle (Physics)

This figure illustrates the shape of a vector in the cartesian coordinate plane.
Figure 2: Vector

The properties of the unit circle are used in physics for decomposing vectors into their components. For example, say we want to calculate the components of a velocity vector given its speed and direction. The trigonometry of the unit circle allows us to decompose the vector into its horizontal velocity using its speed (magnitude) denoted and direction (angle) . This is shown below.

Unit Circle (Linear Algebra)

The trigonometry of the unit circle is used in linear algebra for transformation matrices. For example, the matrix above transforms the vector by the angle (theta). Futhermore, because the values are scalar in the range from to , the matrices also inherit some useful properties.

Unit Circle (Complex Plane)

Finally, the unit circle also appears in the plane of complex numbers in the famous formula: Euler’s Formula. Given an angle the formula returns the point on the unit circle in the complex plane corresponding to . This is shown below.

Euler's Formula in the Complex Plane
Figure 3: Euler's Formula

Summary

Hopefully this page gives you an idea of some of the applciations of the unit circle. I will keep updating it with examples over time.

Links