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 θ represents the current angle corresponding to the right triangle with an adjacent side (x) and an opposite side (y).

Notation

Right Triangle Unit Circle Notation

To make things a little more clear, let’s look at the angle and right-triangle separately. By convention, angles are measured starting from the positive right direction and the unit of measure is radians. The right triangle has a hypotenuse of one and is formed by a point along the perimeter of the unit circle.

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 one, so cosine corresponds directly with the x-component. This is shown in the interactive below:

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 one, so sine corresponds directly with the y-component. This is shown in the interactive below:

Unit Circle
Cosine
cos(θ)
Sine
sin(θ)
Unit Circle Chart
Memorize Unit Circle Chart