Re: Sin Cos Tan, why not Sin Sec Tan?

From: James Dolan (jdolan_at_math-cl-n03.math.ucr.edu)
Date: 07/15/04 

Date: Thu, 15 Jul 2004 22:41:17 +0000 (UTC)

in article <nomjc.1663$K53.873@news-server.bigpond.net.au>,
cassandra thompson <cass.harley@bigpond.com> wrote:

|I am learning trigonometry in preperation for actually teaching it.
|I am enjoying it, and would like to think I am getting a good
|understanding, however I am unsure about hte following.
|
|When talking about highschool level trigonometry we often use
|'SOHCAHTOA' as a way to remember that:
|Sin@ = O/H
|Cos@ = A/H
|Tan@ = O/A
|
|Further on we learn that 3 other functions exist that are the inversion
|of the first three
|
|CSC@ = H/O
|SEC@ = H/A
|COT@ = A/O
|
|So that Sin@ = 1/CSC@
| Cos@ = 1/SEC@
| Tan@ = 1/COT@
|
|
|My question is why is the cofunction of Sin, ie Cosine placed in the
|first three that are learnt. Wouldn't it make more sense to group them as
|
|Sin@ = O/H
|Sec@ = H/A
|Tan@ = O/A
|
|Then introduce the cofunctions
|Cos@ = A/H
|CSC@ = H/O
|Cot@ = A/O
|
|This seems alot more clear to me. Is there some mathematical reason that
|I am missing?

the name "trigonometry" suggests that the subject is all about
triangles, but that's very misleading; what trigonometry secretly
_really_ is is the study of the points on the unit circle. (the right
triangles that show up are just auxiliary devices used to highlight
the points on the unit circle.) from this point of view it's pretty
clear why cosine and sine are the crucial variables; they're the x and
y coordinates of the point on the unit circle.

so in fact besides not bothering with secant and cosecant and
cotangent, it's probably better not to bother with tangent either.
and cosine should generally come before sine, of course, since x comes
before y after all.

of course this tends to reduce trigonometry to about a five-minute
lesson, but that's probably about what it's worth. 

-- 
[e-mail address jdolan@math.ucr.edu]

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