Interactive Unit Circle

This interactive illustrates the connection between the unit circle and the trigonometric functions cosine, sine and tangent. Click and drag the two control points to change the angle or press the animate button. The symbol represents the current angle corresponding to the right triangle with an adjacent side represented by and an opposite side represented by . Note, by convention angles are measured using radians.

Explanation

The unit circle is a unit of radius placed at the center of the Coordinate System. A point along the perimeter of the circle naturally has the geometry of the right triangle.

Unit Circle Notation
Right Triangle

There are two equivalent ways to define the geometry of the right-triangle on the unit circle. 1) The right triangle is formed by the coordinate lengths which define the adjacent and opposite lengths of the triangle. 2) The right triangle is formed by the hypotenuse of length and the angle . These are explored in the two interactives below, click and drag either of the blue control points.

Note, by convention angles are measured using the radians angle system. All angles start in the positive x-direction with the counter-clockwise representing the positive direction. A full rotation in radians is equal to (tau) radians which is approximately radians.

Sine Function

Definition of the sine Function

The sine function is defined as the ratio of the right-triangle’s opposite side over the hypotenuse. In the case of the unit circle, the hypotenuse is of length , so when given an angle the sine function returns a value which corresponds directly with the length of the opposite side. This is shown in the interactive below, where the input angle is highlighted in red and the output of the function is highlighted in blue:

Cosine Function

Definition of the Cosine Function

The cosine function is defined as the ratio of the right-triangle’s adjacent side over the hypotenuse. In the case of the unit circle, the hypotenuse is of length , so when given an angle the cosine function returns a value which corresponds directly with the length of the adjacent side. This is shown in the interactive below, where the input angle is highlighted in red and the output of the function is highlighted in blue:

Broader Applications

The connections formed on the unit circle can easily be scaled to broader applications. For example, coordinates defined in the Cartesian and Polar systems can be transformed from one to the other using the geometry defined on the unit circle.

Related Links

Unit Circle | Concept

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.

Unit Circle Chart | Concept

The unit circle chart shows the position of the points along the circle that are formed by dividing the circle into eight and twelve parts.

Radians | Concept

Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

Right Triangle | Concept

A right triangle is a triangle where one of the three angles is a perpendicular angle. There are three sides of the right triangle: the adjacent, opposite, and hypotenuse sides.

Cosine | Function

The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the right-triangle's adjacent side over its hypotenuse.

Sine | Function

The sine function returns the sine of a number provided in radians. In geometric terms, given the angle of a right-triangle as input, the function returns the ratio of the triangle's opposite side over its hypotenuse.

Tangent | Function

Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.