**ERIC Number:**EJ1044597

**Record Type:**Journal

**Publication Date:**2014-Sep

**Pages:**6

**Abstractor:**ERIC

**Reference Count:**N/A

**ISBN:**N/A

**ISSN:**ISSN-0025-5769

Unit Circles and Inverse Trigonometric Functions

Barrera, Azael

Mathematics Teacher, v108 n2 p114-119 Sep 2014

Historical accounts of trigonometry refer to the works of many Indian and Arab astronomers on the origin of the trigonometric functions as we know them now, in particular Abu al-Wafa (ca. 980 CE), who determined and named all known trigonometric functions from segments constructed on a regular circle and later on a unit circle (Moussa 2011; Bressoud 2010). Determinations of the trigonometric function values from right-triangle ratios came later, with Johann Muller Regiomontanus's work "De Triangulis Omnimodis" of 1533 (Van Brummelen 2009) and others. Recently scholars have advocated teaching trigonometry using the unit circle within the Greek and the Arabic- Indian historical context (Bressoud 2010). In Azael Barrera's trigonometry class, his students learn to determine all trigonometric functions using only the unit circle. The unit circle construct that he uses (described herein) was adapted from Abu al-Wafa (Moussa 2011) and was later revived by Marolois (1627). In this article, Barrera describes how students first draw the ray of a given angle on a unit circle in the standard position using a protractor; then they draw the segments that intersect the angle's ray. Finally, they measure the lengths of the six segments formed. Signs of the functions in the different quadrants in the unit circle are determined using vertical tangent lines. After finding all trigonometric functions, the students are asked to check their results by using a calculator. To facilitate their measurements, Barrera's students use unit circle worksheets that are either unmarked or marked in degrees or radians. Next, students learn to determine the values of the inverse trigonometric functions directly from the unit circle. To do so, he designed the method detailed here based exclusively on the unit circle. A bibliography is included.

Descriptors: Mathematics Instruction, Trigonometry, Mathematical Concepts, Measurement, Worksheets, College Mathematics, Misconceptions, Teaching Methods

National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. Tel: 800-235-7566; Tel: 703-620-3702; Fax: 703-476-2970; e-mail: orders@nctm.org; Web site: http://www.nctm.org/publications/

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education; Postsecondary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A