# 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 related pages below.

## Unit Circle Trigonometry

The properties of a point on the unit circle form the input and output of the trigonometric functions. The trigonometric functions are often introduced using the geometry of the right triangle as shown on this page and then extended using the circle definitions shown below.

Each of the functions takes in an angle measured in radians as input and returns a ratio as output. As shown above, the defintions can be visualized by the circle defined by the point, where forms the shape of the input angle (theta).

The significance of the unit circle and its radius of length is that the properties of the point on the unit circle defined by the angle correspond exactly to the output of the trigonometric functions. This can be seen by substituting into the circle defintions above.

Since the angle (theta) is measures in radians, the arc-length of the angle is also exactly equal to the measured angle.

## Trigonometric Functions

The trigonometric functions: sine, cosine, and tangent are periodic wave functions that repeat forever. The functions use radian angles as input because radians give the functions useful properties. The radian angle system is visualized below using the circle constant (tau) which represents a full rotation in radians.

### Sine

Given an angle in radians, sine returns the vertical coordinate of the point on the unit circle corresponding to the angle. This is shown in the interactive below which highlights the input and output of the function on the unit circle and the plot of sine. Click and drag either of the blue points.

### Cosine

Given an angle in radians, cosine returns the horizontal coordinate of the point on the unit circle corresponding to the angle. This is shown in the interactive below which highlights the input and output of the function on the unit circle and the plot of cosine. Click and drag either of the blue points.

### Tangent

Given an angle in radians, tangent returns the length of the line tangent to the point on the unit circle corresponding to the angle. This is shown in the interactive below which highlights the input and output of the function on the unit circle and the plot of tangent. Click and drag either of the blue points.

## Summary

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.

The property each trigonometric function corresponds to on the unit circle is shown in the figure above and summarized in the table below.

Geometry Function Sine Function
The output of sine corresponds to the distance from a point on the perimeter of the unit circle to the -axis. Cosine Function
The output of cosine corresponds to the distance from a point on the perimeter of the unit circle to the -axis. 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. 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.