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.

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
References
 McAdams, David E.. All Math Words Dictionary, unit circle. 2nd Classroom edition 201501084799968. pg 186. Life is a Story Problem LLC. January 8, 2015. Buy the book
 unit circle. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 12/17/2018. http://www.merriamwebster.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.)