For me,all trigs(sin,cos,tan,sec,cosec,cot) could be replaced by one,say,sin:
cos=sqrt(1-sin2)
sin,cos,sin/cos, 1/cos,1/sin,cos/sin

Any time you see
xxx/cos(a)
,
that''s xxx*sec(a) , and 1/cos form was used just because it''s sometimes hard to remember what''s sec are %).So,if you don''t see secant too often,that only because you don''t see secant in xxx/cos(a) . Probably your teacher too. Say,1/sin (==cosec) are used in SLERPS. It''s like not seeing x-0.5 in 1/sqrt(x)

Probably we have special symbols for sin/cos, 1/cos,1/sin,cos/sin
because it''s not harder to calculate it than single sin or cos,and sin or cos could be as well derived from any other trigonometric function.So there''s no reason to derive everything from sin and cos.

It''s like inverse square root in modern processors.It''s faster to calculate inverse square root than square root %-).

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Yet another reason, when sin-1b are mixed with sin2a ,logically,sin-1b=1/sin b OR sin2a = sin(sin (a))
so cosecant saves from confusion that sin-1b=1/sin b because it could be shorter written as csc b

All six trigonometric functions have geometric interpretations.

http://aleph0.clarku.edu/~djoyce/java/trig/

Although not labelled in that applet, the length of the segment that joins the origin and the point (1,tan x) is the secant, and the length of the segment that joins the origin and the point (cotan x, 1) is the cosecant.

That's why I think that it is natural to have names for them.

[edited by - Alvaro on February 28, 2004 11:10:01 AM]

Yes,as natural as name for tangent (sin/cos)

Although initially defined in the geometric context, sine and cosine also happen to be useful as basis for periodic functions (Fourier series), and appear very naturally in solutions to differential equations. In the end that''s all because exp(i*x)=cos(x)+i*sin(x).

The other trigonometric functions don''t appear as naturally, so their application is mostly limited to their original geometric context.

quote:Original post by Emonious
Thanks guys, but I guess what I''m trying to ask is this. I''ve seen waves similar to cosine and sin as radio waves or other applications to things in real life. Even tan I''ve seen show up somewhere, but I can''t remember where... But Csc and Sec, I have seen those no where, and I can''t seem to figure out where they would apply?

I suppose you don''t see the need for making substitutions when doing integration, either? Or the hyperbolic trig functions? In a lot of contexts they don''t add anything to the domain, but they can make things much easier for humans to work with. "But Mathematica/Maple/Matlab/whatever can do all that for me!" Yes, they do a lot, but can''t do everything. Quite often the form they''ve returned hasn''t been very useful for the problem at hand, and I''ve had to perform extra algebra on the result to get something usable (What''s a simplification in one context can be a complification in another). Also, we don''t use math exclusively for talking with computers, we also use it for talking with other people. sec and csc can make things much easier for humans to understand once you''re used to the notation.

For what its worth, I''m majoring in math and physics. We use trig quite a lot so these "alternate" forms are useful (both for the simplification of formulas and their geometric interpretation). I don''t know how common trig is in your field (I have no formal training in computer science). If it''s not very common, then I can imagine 1/cos(t) being much more readable than sec(t) and would recommend you stick to the former (if only to keep communication clearer).