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Teachers' Lounge Discussion: Geometry or Trig first?

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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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