ArcCot--relationship with ArcTan
Chen Yaohan
inverse cotangent function with someone, based on what math we have learned.
(A) arccot x = Pi/2 - arctan x
(B) arccot x = arctan 1/x
My calculus book used the definition A when it talked about the derivatives
of inverse trig functions. I think it said that definition B is also used in
some situations, but I don't have the book now.
It's obvious why my calculus book and most calculus study guides found on
the Internet use the definition A--arccot would be continuous on its domain,
and the complementary angle relationship is preserved. But Mathematica uses
definition B, or a definition closer to B. See
http://mathworld.wolfram.com/InverseCotangent.html
So, is there any merit of definition B, besides "convenience"? Should we
sacrifice the continuity of inverse cotangent just to satisfy the simple
"x-reciprocal" relationship?
And this page
http://mathworld.wolfram.com/InverseTrigonometricFunctions.html
says the "domain" (my math teacher would say range) of ArcCot is (0, Pi/2)
or (-Pi, -Pi/2). Where does that come from? It's not even consistent with
the graph on Wolfram's own inverse cotangent page! The range for ArcCsc
listed there is strange too. Also, Wolfram doesn't say that arccot is not
differentiable because of discontinuity at x=0 when it lists the first
derivative of ArcCot[z], while definition B would require that. Is it
something in more advanced mathematics that I don't know yet?
When I argued over the two definitions few weeks ago, my opinion was that
definition B was acceptable, or even should be authoritative, because
Wolfram and possibly other systems use it. But after my opponent pointed out that
inconsistency, I didn't know what to say. Still, I believe there is a reason
why definition B not only survived but also is used in these authoritative
software. Could anybody explain to me? Thanks!
Yaohan Chen
David W. Cantrell
> I argued over the more appropriate definition of (or algorithm for) the
> inverse cotangent function with someone, based on what math we have
> learned.
Such arguments could probably be continued _ad infinitum_!
> (A) arccot x = Pi/2 - arctan x
> (B) arccot x = arctan 1/x
>
> My calculus book used the definition A when it talked about the
> derivatives of inverse trig functions. I think it said that definition B
> is also used in some situations, but I don't have the book now.
>
> It's obvious why my calculus book and most calculus study guides found on
> the Internet use the definition A--arccot would be continuous on its
> domain, and the complementary angle relationship is preserved. But
> Mathematica uses definition B, or a definition closer to B. See
> http://mathworld.wolfram.com/InverseCotangent.html
Right: Closer to, but not the same as, B. It is not the same as B
because, in Mathematica, ArcCot[0] yields Pi/2, while ArcTan[1/0]
yields Indeterminate. FWIW, however, note that FullSimplify[ArcTan[1/x]]
does give ArcCot[x].
> So, is there any merit of definition B, besides "convenience"? Should we
> sacrifice the continuity of inverse cotangent just to satisfy the simple
> "x-reciprocal" relationship?
Since I favor definition A (at least for the purposes of real analysis),
I'll leave this for proponents of Mathematica's definition. Just realize
that either definition is "correct". [BTW, although it's probably a bit
above your level at present, you might be interested to look for
information about "branch cuts".]
> And this page
> http://mathworld.wolfram.com/InverseTrigonometricFunctions.html
> says the "domain" (my math teacher would say range) of ArcCot is (0,
> Pi/2) or (-Pi, -Pi/2). Where does that come from? It's not even
> consistent with the graph on Wolfram's own inverse cotangent page!
Right you are! Almost undoubtedly, the two MathWorld entries which
you cited were written by Eric _long ago_. Many of those early entries
had errors. And, as you see, some of those errors have yet to be
corrected. [I'll be sending an expanded version of this message to Eric,
and so these errors will soon be eliminated.]
Anyway, you're absolutely correct that the table should say Range,
instead of Domain (because the first column is talking about the
inverse functions, rather than the original trig functions). Furthermore,
if you wish to take the reals as the domain of the inverse cotangent,
then the range which is appropriate for Mathematica's definition is
-Pi/2 < y < 0 or 0 < y <= Pi/2
and the range which is appropriate using definition A would be
0 < y < Pi.
> The range for ArcCsc listed there is strange too.
Correct again. It's the same sort of error. If the domain of the inverse
cosecant is taken to be a subset of the reals, then the range should,
to correspond with Mathematica's definition, be
-Pi/2 <= y < 0 or 0 < y <= Pi/2.
> Also, Wolfram doesn't say
> that arccot is not differentiable because of discontinuity at x=0 when it
> lists the first derivative of ArcCot[z], while definition B would require
> that.
It is not a good idea to expect _any_ computer algebra system
to say such things!
The following might (or might not) amuse you:
In[12]:= D[Floor[x],x]
Out[12]= Floor'[x]
In[13]:= %/.x->1
Out[13]= Floor'[1]
In[14]:= N[%]
Out[14]= 12.5946
Regards,
David
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