Sine and SecantDate: 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/ |
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