Arcsin, Arccos, Arcsec Are Confusing in their RangesDate: 12/01/2011 at 19:54:08 From: jayson Subject: domain of arcsec Is the domain for arcsin 0 to pi? For arccos, is it -pi/2 to pi/2? I'm confused about how these are determined. I think the domain of arcsin is 0 to pi because y is positive at these values. Is that the right reason? Also, what is the domain of arcsec? Date: 12/01/2011 at 23:33:39 From: Doctor Peterson Subject: Re: domain of arcsec Hi, Jayson. When you say "domain," you really mean "range," right? The restricted domain used for the secant, etc., before taking the inverse, becomes the range of the inverse function. This range is not really "determined" as if we just have to study the function to find what its range MUST be. Rather, we make a partially arbitrary choice of a domain to which we can restrict the trig function -- a choice that will yield all possible values of the function exactly once, and be as well-behaved as possible. This is described here: Inverses of Trigonometric Functions http://mathforum.org/library/drmath/view/61051.html The general goal is to pick a range that is complete, contiguous (or as nearly so as possible), close to zero, and preferably positive rather than negative. For the sine, we can accomplish all but the last by using [-pi/2, pi/2], where sin(x) goes from -1 to 1. For the cosine, that wouldn't be one-to-one or give all possible values. But the next best choice, [0, pi], does meet all the goals. The tangent works the same as the sine, except that you can't include -pi/2 or pi/2. When you get to the cotangent, secant, and cosecant, the right choice is not quite as obvious. That's especially true of the cotangent. At first you'd think the cotangent should obviously have the same domain as the cosine (without the end points), much as we do for the tangent with regard to the sine; that makes its graph continuous, and that is the usual choice. But, as I said, nothing forces us to make that choice, and there are some reasons in favor of instead choosing [-pi/2,0) U (0,pi/2], even though that is not a contiguous interval. Here's a nice explanation I found (which happens to quote me!): http://www.squarecirclez.com/blog/which-is-the-correct-graph-of-arccot- x/6009 Now, in my quoted answer, I was not so much stating a strong opinion on this choice as I was giving a reason for the choice I was asked about, which happens to agree with our FAQ. My explanation for the contiguous range seems reasonable; why in the world would anyone choose the other? The following site states the convention chosen by Mathematica software: http://mathworld.wolfram.com/InverseCotangent.html There are at least two possible conventions for defining the inverse cotangent. This work follows the convention of Abramowitz and Stegun (1972, p. 79) and Mathematica, taking cot^(-1)x to have range (-pi/2,pi/2], a discontinuity at x = 0, ... This definition is also consistent, as it must be, with Mathematica's definition of ArcTan, so ArcCot[z] is equal to ArcTan[1/z]. A different but common convention (e.g., Zwillinger 1995, p. 466; Bronshtein and Semendyayev, 1997, p. 70; Jeffrey 2000, p. 125) defines the range of cot^(-1)x as (0,pi), thus giving a function that is continuous on the real line R. Extreme care should be taken where examining identities involving inverse trigonometric functions, since their range of applicability or precise form may differ depending on the convention being used. So the reason for their choice is that they want it to be always true that arccot(x) = arctan(1/x) This makes sense, since cot(x) = 1/tan(x) This implies that arccot and arctan have to have the same range! Looking at lists of identities in a site that uses the more common continuous range ... http://en.wikipedia.org/wiki/Inverse_trigonometric_functions ... we find that their equivalent identity is arctan(1/x) = pi/2 - arctan(x) = arccot(x), if x > 0 The first equality here amounts to the cofunction identity, that cot(x) = tan(pi/2 - x) They can say it is equal to arccot(x) only for positive x, because for negative x, their arccot would be positive while their arctan would be negative. (Our own Formula FAQ does the same thing.) If x < 0, you would have to say arctan(1/x) = arccot(x) - pi, x < 0 I don't find as much disagreement about the range of arcsec and arccsc. Here, the usual restricted domain is [0,pi/2) U (pi/2,pi] for secant, and [-pi/2,0) U (0,pi/2] for cosecant. These follow my guidelines above, the former matching cosine and the latter matching sine. This makes their reciprocal identity work nicely, too. The following page mentions the reason for an alternative choice, based on calculus: Differing Definitions of arcsec(x) Lead to Confusion over Signs http://mathforum.org/library/drmath/view/69193.html Ultimately, the answer to your question is (a) it's a choice, so to find what the range of an inverse trig function is, you just have to look in your text to see what they're using; and (b) the choice is made based on what will make the kind of math you're doing (trig identities, calculus, etc.) work best. That's true of a lot of definitions in math: convenience rules! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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