Exact Values of Trigonometric Functions

Pronunciation: /ɪgˈzækt ˈvæl.juz ʌv ˌtrɪg.əˈnɒm.ɪ.trik ˈfʌŋk.ʃənz/ Explain

The exact values of trigonometric functions are values of trigonometric functions of certain angles that can be expressed exactly using expressions containing real numbers and roots of real numbers. These values are also called analytic values of trigonometric functions or trigonometric values of special angles. These angles are often used on tests where a calculator is not allowed, so it is useful for students to memorize these angles and the values of trigonometric functions of these angles. A mnemonic for remembering these values is at the end of this article.

Exact Values of Trigonometric Functions
0 rad000
30°3pi/2 rad1/2square root(3)/2square root(3)/3
45°pi/4 radsquare root(2)/2square root(2)/20
60°pi/3 radsquare root(3)/21/2square root(3)
90°pi/2 rad00undefined
180°pi rad0-10
270°3*pi/2 rad-10undefined
Table 1

Mnemonic What is this?

A mnemonic for remembering the values of sine for special angles is:

sin(0)=(1/2)*square root(0); sin(30)=(1/2)*square root(1); sin(45)=(1/2)*square root(2); sin(60)=(1/2)*square root(3); sin(90)=(1/2)*square root(4);
Figure 1: Mnemonic for sine of special angles

A mnemonic for the cosine values of special angles is similar:

cos(0)=(1/2)*square root(4); cos(30)=(1/2)*square root(3); cos(45)=(1/2)*square root(2); cos(60)=(1/2)*square root(1); cos(90)=(1/2)*square root(0);
Figure 1: Mnemonic for cosine of special angles


  1. McAdams, David E.. All Math Words Dictionary, exact values of trigonometric functions. 2nd Classroom edition 20150108-4799968. pg 74. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Exact Values of Trigonometric Functions. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/e/exactvaltrig.html.

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Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/1/2010: Added "References". (McAdams, David E.)
12/26/2009: Combined with Analytic Values of Trigonometric Functions and added reference. (McAdams, David E.)
1/7/2009: Initial version. (McAdams, David E.)

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