Re: Sin Cos Tan, why not Sin Sec Tan?
From: George W. Cherry (GWCherryHatesGreenEggsAndSpam_at_alum.mit.edu)
Date: 07/16/04
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Date: Fri, 16 Jul 2004 16:57:05 GMT
"Bill" <bill.thomson@tka.co.za> wrote in message
news:4b1e845e.0407160003.5d2229ca@posting.google.com...
> jdolan@math-cl-n03.math.ucr.edu (James Dolan) wrote in message
news:<cd716d$sji$1@glue.ucr.edu>...
> > in article <nomjc.1663$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.
>
> Your sine and cos description is very succinctly put.
>
> But I have always wondered about Tan - why was this not defined as a
> tangential intersect of x and y (a tangent to the unitary circle at 45
> degree would intersect x and y at 1.414 (or sqrt2 whichever you
> prefer))?
What is the y intersect of the tangent to the
unit circle when x = 1? What is the x intersect
of the tangent to the unit circle when y = 1?
What would be the benefit of your idea?
> Would this not have been more logical?
Why "more logical"?
> OK so we would have to have two
> definitions of Tan - say a sinTan and a cosTan - the values of these
> 'Tans' will be the reciprocal of their sin and cos counterparts.
>
> Kind of looking at x and y inside the circle (sin and cos) compared to
> looking outside the circle (sinTan and cosTan). Just think of all the
> nice formula fiddling that can take place.
>
> Bill
- Next message: David M Einstein: "Re: Length of sequence of consecutive primes starting at 2"
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- In reply to: Bill: "Re: Sin Cos Tan, why not Sin Sec Tan?"
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