# Unit Circle

A unit circle is a circle with a radius of 1. It is used frequently in the context of trigonometry, with a unit circle centered at the origin, (0, 0), and angles described between 0° and 360°, or 0 and 2π radians.

When a ray is drawn from the origin of the unit circle, it will intersect the unit circle at a point (x, y) and form a right triangle with the x-axis, as shown above. The hypotenuse of the right triangle formed will always be 1. Based on the Pythagorean theorem,

x^{2} + y^{2} = 1^{2}

or, after simplifying,

x^{2} + y^{2} = 1

It is important to note that this holds true for all points on the unit circle, not just those in the first quadrant.

## Unit circle definitions of trigonometric functions

The unit circle is often used in the definition of trigonometric functions. In a plane with a unit circle centered at the origin of a coordinate system, a ray from the origin forms an angle, θ, with the x-axis. The ray intersects the unit circle at point A. The tangent line to the unit circle at point A intersects the x-axis at point B and intersects the y-axis at point C. The y-coordinate of the point A is the sine (or sin) value of θ. The x-coordinate of the point A is the cosine (or cos) value. The line segment BC is tangent to the unit circle at point A. The length of AB is the tangent (or tan) value. The length of AC is the cotangent (or cot) value. The x-coordinate of the point B is the secant (or sec) value. The y-coordinate of the point C is the cosecant (or csc) value. The definitions are noted in different colors on the diagram below.

Given cos(θ) and sin(θ) are the coordinate values of point A, x=cos(θ) and y=sin(θ), and using the Pythagorean theorem, we can deduce an identity that is commonly used in trigonometry:

cos^{2}(θ) + sin^{2}(θ) = 1

## Commonly used angles

There are some values on the unit circle worth remembering because of how frequently used they are in trigonometry. These angles are 30°, 45°, and 60°. In radians, these correspond to , , and . Below is a table of these values, as well as a figure of the values on a unit circle. Note that the values for tangent can be found using so it is not necessary to memorize the tangent values.

Angle | sin(θ) | cos(θ) | tan(θ) |
---|---|---|---|

1 | |||

As can be seen from the table and from the figure above, for the most part, there are three values to remember: , , and . Because of the nature of the unit circle, all the values are the same for their respective angles in the unit circle, with the only difference being the sign based on the quadrant. Therefore, remembering these three values and how they correspond to the multiples of 30°, 45° and 60° would enable you to fill in all the other values in the unit circle above.

0° or 0 (which has equivalent sine and cosine values as 360° or 2π), 90° or , 180° or π, and 270° or are other values that should be remembered. At any of those angles, either sin(θ) or cos(θ) is equal to 0, and the value that is non-zero is equal to 1 or -1 depending on which quadrant the angle is in.

Angle | sin(θ) | cos(θ) | tan(θ) |
---|---|---|---|

0 | 1 | 0 | |

1 | 0 | Undefined | |

180° or π | 0 | -1 | 0 |

-1 | 0 | Undefined |

## Remembering common values

There are various ways to remember the common values described above. If you are able to purely memorize the values for sine and cosine, it is always possible to calculate the tangent values. If not, the method described below may make it easier for you to remember the values.

The values for sine and cosine correspond to each other. At 45°, the values for both sine and cosine are equal. At 60°, the value of sine is equal to the value of cosine at 30°, and vice versa.

This can be seen if you remember the value of sine as the numerator counting up from one, as 1, 2, 3, and cosine as counting down from three, as 3, 2, 1. Then take the square root of the numerator, and remember that the denominator of the values for both sine and cosine for the three chosen angles (and their multiples) is always 2.

For cosine, we count down instead, starting from 3:

It may also help to see the values written together, as in the unit circle:

See also sine, cosine, tangent.