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Emonious

Uses of secant and cosecant?

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During my path for a major in computer science I've crossed trigonometry lately on the long way up to physics 4a, so this question's answer may seem obvious to some of you who are mathmatical gods. While I can understand the value of cos, sin and almost tan, I can't for the life of me figure out what value sec and csc are. Even when graphed they seem to make even less sense. I've asked my college teacher this question, but she didn't have a clue either. My guess is electronics and such more than any real work equations, but that is as good as I can come up with. Any of you have deep knowledge and maybe even experience in using sec and csc in your field or study? I'm hoping to find a bit of information out just to understand the reason for learning this material However, basic it might seem, just learning it "because" is never quite good enough for me. I want to know the reason Thanks to any who can help. [edited by - emonious on February 28, 2004 1:05:00 AM]

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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 %-).

**************************************
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

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You could just view it as this:

sec = 1/cos
csc = 1/sin

So anytime you need 1/cos, throw in sec instead. It''s just manipulation, really.

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I don''t think they have any "real" use (anything that can be done with the secant/cosecant can also be done with the cosine/sine)... But they do make certain identities look prettier, e.g tan^2(x) + 1 = sec^2(x).

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In the days before calculators and computers, people had to use look up tables. These often came with the secant and cosecant values, presumably so a sliderule could be updated directly with 1/sin etc.

Just a guess, not a history lesson!



Stevie

Don''t follow me, I''m lost.

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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]

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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?

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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.

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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).

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