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

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 07/15/04 

Date: Thu, 15 Jul 2004 00:53:34 -0600

In article <YPoJc.1781$K53.825@news-server.bigpond.net.au>,
 Cassandra Thompson <cass.harley@bigpond.com> wrote:

> Jeremy Targett wrote:
>
> > Cassandra Thompson <cass.harley@bigpond.com> wrote:
> >
> >
> >>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
> >
> >
> >>My question is why is the cofunction of Sin, ie Cosine placed in the
> >>first three that are learnt.
> >
> >
> > For your students: cos and sin are often used to find the lengths of the
> > non-hypoteneuse sides of a right-angled triangle, given the angle. This
> > happens for example when decomposing a vector of length x at angle @ from
> > the coordinate axes: one component is xsin@, and the other is xcos@.
> > There's a symmetry there that you wouldn't get if you used one of the
> > other functions. Likewise in identities like sin^2(@) + cos^2(@) = 1. Or
> > if you introduce the functions on a circle, you'll show the kids that the
> > sine of an arc-length in radians gives its vertical component and the
> > cosine its horizontal component. Again, there's a symmetry between the two
> > that it would be crazy to change. If you do lots more work with these
> > functions it should become clear to you that sin and cos are a natural
> > pair, whereas the relationship between sin and sec, for example, is more
> > obscure. Then you'll be ready to teach the kids. Good luck to all
> > involved.
> Thanks, that does make it a bit clearer.
> I can see how sin and cos are a natural pair, being that cosine is the
> cofunction of sin.
>
> I am still trying to understand why we have a tendency (as teachers) to
> present the three as if there were an inpenetrable group.
> ie (sin, cos, tan)
>
> When it appears, at least to me, that we are really trying to teach them:
> (sin and its cofunction, cos) and (tan).
>
> (Sec, and its cofunction csc) and (cot) get a mention one the student
> fully understands the first three.
>
> [I hope the brackets indicate how my mind groups these functions).
>
> Would it be just as worthwhile to the student to teach instead.
> (sec and its cofunction csc), and tan???
>
> I know this question is more about pedagogy then mathematics, it is
> important for me to understand why we teach in the way that we do. I
> will continue working through more trigonometry, however I imagine that
> it would be in the high levels of trig that I would eventually see why
> sin and cos are more important then sec and csc. At the moment I really
> see them as being very equal, and can't understand why one pair is
> better then the other.

Actually, for a right triangle with unequal shorter sides, there are six
ratios of 2 different sides possible, and each of them has a name. If
this is pointed out at the beginning and it is explained that eventually
each of the six will be considered, with suitable mnemonic help to keep
them straight, then there is a good deal less confusion among students
newly presented with these six "ratio" functions.

Use complimentary angles: the compliment of an angle, co-angle, is a
right angle minus the original angle, which for either acute angle in a
right triangle means the other acute angle. Any function of the co-angle
is the co-function of the original angle, which cuts it down to 3
functions and the co-relations. Sine, cosine, secant and cosecant
involve the hypotenuese, but the tangent and cotangent do not.

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