Teacher2Teacher 
Q&A #304 
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From: Richard Askey <askey@math.wisc.edu> To: Teacher2Teacher Public Discussion Date: 1999013100:19:55 Subject: Re: Geometry or Trig first? Trigonometry depends on a few facts from geometry which students should understand before starting to learn trigonometry. The first is similarity, so that students understand why the ratios of corresponding sides of similar triangles are the same. Without this, it is not possible to define the trig functions. The second fact is the Pythagorean theorem. Students should understand why this is true, and that is not trivial. I do not know anyone who thinks the Pythagorean theorem is obvious. Knowing at least a couple of ways to prove this gives some insight into why it is true. The Pythagorean theorem requires that the sum of the angles in a triangle be 180 degrees, and this is equivalent to the parallel postulate. All of this takes some time to learn, and without it, the foundation on which trigonometry is build is too fragile for it to stick. Finally, to derive the law of sines, the law of cosines and the addition formulas for sines and cosines, it is necessary to know a bit more. One can get away with the simple fact that any triangle can be decomposed into the union of two right triangles, but that is not the usual way of deriving the addition formulas. A new trig book by I.M. Gelfand and Mark Saul will be published by Birkhauser which has proofs of these facts by decomposition, and of the addition formulas by other methods. The usual way in texts now is to use the invariance of the unit circle under rotation. In any case, geometry is a very important part of mathematics, and trigonometry meshes algebra and geometry. Take geometry before trigonometry.
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