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2011 AAPT Summer Meeting
July 30-August 3, 2011
Omaha, Nebraska

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FF01:

The Law of Refraction without Trigonometry: Beaten to it by Descartes!
  Location: SS Ballroom DE
  Date: Wednesday, Aug.3
  Time: 8:00 AM -8:10 AM
  Author: David Schuster, Western Michigan University
269 387-5844, david.schuster@wmich.edu
  Co-Author(s): Betty Adams , Adriana Undreiu
  Abstract: Investigating and discovering a law for refraction is potentially an ideal activity for inquiry-based physics. However, the law of refraction involves sine functions; this complicates an empirical search for a law (as it did historically) and may also seem to preclude it for students with no trigonometry. Wanting a guided-discovery approach nonetheless, we "invented" a geometrical representation: incident and refracted ray directions can be specified not only by angle but by semi-chords in a reference circle. This proves very successful: students discover that various possible relationships, such as angle ratios, are initially promising but do not work at large angles; and they finally arrive at a simple and visually elegant law: the ratio of semi-chords for incident and refracted rays is constant. We then found that we had been beaten to this form of the law by nearly 400 years, by Descartes among others! Thus in the case of refraction, exemplary inquiry pedagogy has a counterpart in history. Note that the approach also reveals the underlying meaning of sine functions and a reason why trigonometry was invented. Students then go on to use the semi-chord representation to solve refraction problems by geometrical construction.
  Footnotes: None
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