# 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

 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).

 Table 1

### References

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

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

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

### Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)