The unit circle is a circle centered at the origin, with a radius of one.

The equation of the unit circle is u² + v² = 1.

Note: To avoid labelling conflicts later, the unit circle is graphed in the u-v plane, rather than the x-y plane.

The unit circle provides a visual way to think about trigonometry and trigonometric functions. The unit circle concept takes any equivalence class of similar right triangles and represents the class using a single triangle with a hypotenuse of one. The triangle is oriented in the coordinate plane with the adjacent side along the x-axis, starting at the origin with angle θ (theta).

Recall that ratios of the lengths of corresponding sides of similar triangles are equal.

In the coordinate plane, θ usually represents either:

• a directed arclength starting from the initial point ( 1, 0 ), with counterclockwise being positive. This arclength can be derived from the ratio of the lengths of two sides of a right triangle and as a result has no dimension. It is sometimes given the unit of radians (rad). Click here for more on radian measure.
• an angle with an initial arm along the positive x-axis, with counterclockwise being positive. Angle measure is given in degrees.

Simple locations along the unit circle are based on quadrantal angles as well as the 45°-45°-90° and 30°-60°-90° triangles.

Typical ways of understanding the unit circle involve partitioning the unit circle into four, eight, twelve or twenty-four congruent parts [starting at ( 1, 0 ), wrapping counter-clockwise about the circle].

• the four part unit circle
• the eight part unit circle
• the twelve part unit circle
• the twenty-four part unit circle

Click here to see an illustrative animation showing these circles are all "equivalent".

The unit circle can be used to:

• evaluate simple trigonometric expressions
• solve simple trigonometric equations.