Unit Circle

Pronunciation: /ˈju.nɪt ˈsɜr.kəl/ Explain

A unit circle is a circle with a radius of exactly 1.[1] The unit circle is used to define the sine and cosine functions.

Unit circle
Figure 1: Unit Circle

Relationship Between Unit Circle and Trigonometric Functions

In right triangle trigonometry, sine is defined as opposite/hypotenuse, and cosine is defined as adjacent/hypotenuse. Looking at the unit circle, the radius is the hypotenuse of a right triangle. The red leg of the right triangle (see manipulative 1) is the adjacent leg. So the cosine is equal to adjacent/hypotenuse. But, in the unit circle, the hypotenuse is always 1. So the cosine is always equal to the length of the adjacent leg of the right triangle. By a similar argument, the sine is always equal to the length of the blue leg, the opposite leg.

Circle Trigonometric Identities

Click on the blue points and drag them to change the figure.

Manipulative 2 - Circle Trigonometry Created with GeoGebra.

The circle trigonometric identities are equations that can be used to define trigonometric functions. Given a circle with a center at the origin, the trigonometric functions are defined using the x coordinate of a point on the circle (x), the y coordinate of a point on the circle (y), the radius of the circle (r), and the angle of rotation from an intersection of the circle and the x-axis(θ).

Circle Trigonometric Identities
Table 1


  1. McAdams, David E.. All Math Words Dictionary, unit circle. 2nd Classroom edition 20150108-4799968. pg 186. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. unit circle. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 12/17/2018. http://www.merriam-webster.com/dictionary/unit circle. Buy the book

More Information

  • McAdams, David E.. Circle. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/c/circle.html.

Cite this article as:

McAdams, David E. Unit Circle. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/u/unitcircle.html.

Image Credits

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/17/2018: Removed broken links, updated license, implemented new markup, implemented new GeoGebra applications. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
1/9/2010: Added "References". (McAdams, David E.)
11/27/2008: Added circle trigonometric identities. (McAdams, David E.)
9/15/2008: Expanded 'More Information'. (McAdams, David E.)
6/15/2008: Added manipulative for relationship between trigonometric functions and the unit circle. ( (McAdams, David E.)
3/11/2008: Changed See Also to match current standards. (McAdams, David E.)
8/18/2007: Initial version. (McAdams, David E.)

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