Unit Circle

A circle of radius one placed at the origin of the coordinate system.
Figure 1: Unit Circle

The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions. See also: the Unit Circle Chart which shows the points corresponding to special angles on the unit circle, the Unit Circle Radians Page which discusses the radians and the unit circle and the Applications of the Unit Circle.

Trigonometry

The unit circle is a special case of the circle whose properties represent the input and output of the trigonometric functions for all real numbers. The trigonometric functions are often introduced and defined using the geometry of the right triangle and then extended using the unit circle. The definitions of the three main functions: cosine, sine and tangent are shown below.

This figure illustrates the right triangle.
Figure 2: Right Triangle

The definitions of the functions are extended to the domain of all real numbers by defining the functions in terms of the geometry of the circle[1]. This is shown by figure 3 which illustrates the geometry of the circle defined by the point in the Polar Coordinate System and Cartesian Coordinate System. The angle the point forms is denoted as (theta) and represents the input to the functions.

This figure illustrates the trigonometric functions defined using the circle.
Figure 3: Circle Functions

The significance of the unit circle and its radius of length is that the lengths on the unit circle correspond directly to the input and output of the trigonometric functions. Substituting into the defintions gives us the relationships shown in the equations below.

Variables associated with the Unit Circle
Figure 4: Unit Circle Notation

The angle, measured in radians, corresponds to the distance traveled along the circumference to the point . The output of the cosine function for the given angle corresponds to the horizontal length of the point. The output of the sine function for the given angle corresponds to the vertical length of the point.

Radians

The arc-length of an angle measured in radians on the unit circle corresponds directly to the value of the angle. Radians are the preferred unit for measuring angles in math as they lead to more succinct and elegant formulas[2]. As a result, the trigonometric functions expect input in radians. The interactive below visualizes angles on the unit circle in radians and annotated using the circle constant (tau).

A full rotation in radians is equal to (tau) radians or approximately radians. The positive direction is the start, positive rotation is in the counter-clockwise direction and negative rotation is in the clockwise direction. In relationship to the trigonometric functions, angle input corresponds to a specific point on the circumference of the unit circle.

Circle Functions

The trigonometric functions: cosine, sine and tangent are a set of periodic wave functions that repeat forever. Each function takes in an angle as input which can be visualized with the corresponding point on the unit circle and returns a geometric relation on the unit circle.

Cosine

Given an angle provided in radians, the cosine function returns the horizontal component of the corresponding point on the unit circle. This is shown in the interactive below which highlights the input and output of the function both on the unit circle shown on the left and the graph of cosine shown on the right. Click and drag either of the blue points to see this relationship.

Sine

Given an angle provided in radians, the sine function returns the vertical component of the corresponding point on the unit circle. This is shown in the interactive below which highlights the input and output of the function both on the unit circle shown on the left and the graph of sine shown on the right. Click and drag either of the blue points to see this relationship.

Tangent

Given an angle provided in radians, the tangent function returns the length of the line tangent to the point on the unit circle. This is shown in the interactive below which highlights the input and output of the function both on the unit circle shown on the left and the graph of sine shown on the right. Click and drag either of the blue points to see this relationship.

Summary

The unit circle’s properties represent the input and output of the trigonometric functions for all real numbers as illustrated in figure 5. This makes it a “jumping off point” for introducting the circle definitions of the trigonometric functions and lays a foundation for the applications of the functions in the rest of math and physics.

The unit circle demonstrates the output of the trigonometric functions sine, cosine and tangent through the shared geometry of the right-triangle defined by a point on the perimeter of the circle.
Figure 5: Unit Circle Trigonometry

Students are often quizzed on two groups of special angles and their corresponding points on the unit circle which demonstrate the trigonometry disccussed in this article. This is shown in figure 6 which illustrate the points formed from dividing the unit circle into parts.

The Unit Circle Divided by 12 measured using the circle constant.
Figure 6: Unit Circle Chart Twelfths τ (tau)

references

  1. Extend Trigonometric Functions Using the Circle
    Wumbo (internal)
  2. Radians Versus Degrees
    Wumbo (internal)