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Arcsin, Arccos, Arcsec Are Confusing in their Ranges

Date: 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 

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!):

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: 

   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 ... 

... 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

  Differing Definitions of arcsec(x) Lead to Confusion over Signs 

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 
Associated Topics:
High School Trigonometry

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