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Conceptions of Angles, Trigonometric Functions, and Inverse Trigonometric Functions in College Textbooks

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Abstract

Using expository text and examples available in 10 college textbooks we identify two conceptions of angles, trigonometric functions, and inverse trigonometric functions that rely on either a static or a dynamic definition of angle. Although the textbooks favor a conception of trigonometric functions that is based on a dynamic conception of angle, they split in their definition of inverse trigonometric functions. We argue that transparency in making explicit how these conceptions can be bridged might be useful in understanding difficulties that emerge when solving problems with inverse trigonometric functions.

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Notes

  1. Two-years colleges are post-secondary institutions in the United States that offer the first two years of a college degree. In addition they offer remediation courses in English and mathematics, vocational certificates, and job training. They also offer courses for community enrichment.

  2. Authors use triangle or ratio interchangeably as they describe this approach. We use prefer ratio, but use the author’s preferred terminology when describing the work.

  3. The verification should be that the argument, \( \frac{\sqrt{2}}{2} \) is a number between -1 and 1. The angle that will be obtained will be between –π/2 and π/2.

  4. Students at 2-years institutions can enroll in courses that may transfer to a 4-years degree program in a different institution.

  5. We excluded examples and sections regarding integration and differentiation of these functions because they were not pertinent to this analysis.

  6. Throughout this section we use the letters P, O, R, and Σ to refer to the elements of the quadruplet that define a conception in Balacheff’s model. We keep Σ which is the original notation proposed by Balacheff.

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Authors and Affiliations

  1. 3119 SEB School of Education, University of Michigan, 610 East University, Ann Arbor, MI, 48109-1259, USA

    Vilma Mesa

  2. Google Haifa, Building 30, MATAM, Advanced Technology Center, PO Box 15096, Haifa, 3190500, Israel

    Bradley Goldstein

Authors
  1. Vilma Mesa
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  2. Bradley Goldstein
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Correspondence to Vilma Mesa.

Additional information

This work has been funded in part by the National Science Foundation CAREER Award DRL 0745474 to Vilma Mesa and by the Undergraduate Research Opportunity Program, Michigan Research Community, at the University of Michigan to Bradley Goldstein. Opinions are those of the authors and do not reflect the views of the foundation. Parts of this work have been presented at the 17th Annual Michigan Research Community Spring Research Symposium conference, April 10, 2013, Ann Arbor, MI and at the Annual Conference of the Psychology of Mathematics Education, North American Chapter, November, 2013, Chicago, IL.

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Mesa, V., Goldstein, B. Conceptions of Angles, Trigonometric Functions, and Inverse Trigonometric Functions in College Textbooks. Int. J. Res. Undergrad. Math. Ed. 3, 338–354 (2017). https://doi.org/10.1007/s40753-016-0042-1

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