Sine and Secant

Date: 04/30/2002 at 22:15:34
From: Vaughn Wassmer
Subject: Sine and Secant

When someone applied the terms sine, cosine, tangent, secant, 
cosecant, and cotangent to the trigonometric functions, why didn't 
they make secant the reciprocal of sine and cosecant the reciprocal of 
cosine instead of the other way around, with cosecant being the 
reciprocal of sine and secant being the reciprocal of cosine? Why 
don't they just do it like they did with cotangent being the 
reciprocal of tangent?  It would make it a lot easier to remember, 
don't you think?

Date: 05/01/2002 at 09:22:32
From: Doctor Peterson
Subject: Re: Sine and Secant

Hi, Vaughn.

If you saw where the names come from, as in

   Origin of the Terms Sine, Cosine, Tangent, etc.
   http://mathforum.org/library/drmath/view/52578.html 

then you know why secant IS called secant. Since there is a reason to 
call it secant, there is no reason to call it the cosecant.

You are supposing that trig functions ought to be named in such a way 
that "co's" are reciprocals of other "co's". But there's no such rule; 
the only general rule is that

    co-f(x) = f(90-x)

(in degrees). That is, the "co-something" is the "something" of the 
complement. That's how tangent and cotangent are related, though they 
also happen to be reciprocals. There is no naming convention that 
indicates which functions are reciprocals. (But I'll suggest below 
that it really is more consistent than you realize.)

The name "secant" refers to its representing the length of the secant 
line OB in Dr. Rick's picture. If we gave the name cosecant to that 
function, then the secant would not be the length of a secant line. 
In fact, there is no line you can draw on that picture, without a lot 
of contortions, that would represent the cosecant; so it is natural 
to give primary names to the functions that do have a simple meaning 
(sine, tangent, and secant), and to name the other three as 
co-functions of those.

What's happening here is that, whereas trigonometry started in 
geometry, with each function having a clear relation to a circle, you 
now look at it just as arbitrary functions and expect the names to 
follow an abstract pattern that tells you what they mean from the name 
alone. If we were to start over and give names to the functions based 
only on their relation to the sine and cosine, we could certainly come 
up with a scheme that would meet your expectations; and it might be 
easier to work with, now that nobody thinks in terms of Latin names 
for parts of a diagram. But historically it makes perfect sense how 
the names are assigned, and it's not really hard to memorize their 
relationships. For one thing, it's convenient that the reciprocal of 
any "non-co" function is a "co" function. I find that actually easier 
to follow than if it were the other way. Here is a diagram that 
illustrates the relationships:

          sin   cos
             \ /
      tan ----+---- cot    Opposites are reciprocals
             / \
          sec   csc

      <--------------->
         complements

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/