Re: Re: ArcCos[]

*To*: mathgroup at smc.vnet.net*Subject*: [mg24798] Re: [mg24793] Re: [mg24730] ArcCos[]*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Sun, 13 Aug 2000 03:16:38 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

I have had a further exchange of messages with Gianluca about this topic which lead to conclude that the problem is simpler than I assumed. Below is an extract from a message I sent to Gianluca: I think now that actually it would be possible to implement this, (although it may not be worth the effort). What one would really need to to is to find a mathematical characterization of the minimal polynomials of algebraic numbers of the form Cos[r*Pi] for rational r. This should not be too hard. Then given a radical expression like Sqrt[2 + Sqrt[2]]/2 one simply applies: In[51]:= RootReduce[Sqrt[2 + Sqrt[2]]/2] Out[51]= 2 4 Root[1 - 8 #1 + 8 #1 & , 4] and then one "looks" at the minimal polynomial. For example, look at my other example (Cos[Pi/15]): In[64]:= RootReduce[((-1 + Sqrt[5]))/8 + (Sqrt[(3*(5 + Sqrt[5]))/2])/4] Out[64]= 2 3 4 Root[1 - 8 #1 - 16 #1 + 8 #1 + 16 #1 & , 4] One can see some very clear regularity, suggesting certain conjectures. By comparison, taking a radical "at random": In[57]:= RootReduce[Sqrt[(1 + Sqrt[7])/4]] Out[57]= 2 4 Root[-3 - 4 #1 + 8 #1 & , 2] This clearly looks quite different. So I now believe a rigorous mathematical solution is possible and very likely well known to number theorists, but perhaps no sufficiently important for Mathematica developers to bother about. Andrzej on 8/10/00 6:33 AM, Andrzej Kozlowski at andrzej at tuins.ac.jp wrote: > on 8/9/00 8:31 AM, Gianluca Gorni at gorni at dimi.uniud.it wrote: > >> >> Hello! >> >> I have just come across one more example that shows how there is >> room for improving Mathematica's trig functions. >> >> With Mathematica 4: >> >> v = Cos[ Pi/8 ] // FunctionExpand gives Sqrt[2 + Sqrt[2]]/2 >> >> Still, neither >> >> ArcCos[v] // FunctionExpand nor ArcCos[v] // FullSimplify >> >> give Pi/8, as I would expect, but just >> >> ArcSec[2/Sqrt[2 + Sqrt[2]]] >> >> %%%%%%%%%%%%%%% >> >> An unrelated problem: the following instructions consistently crash >> my Mac Mathematica 4 kernel: >> >> a = Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1]; >> b = D[a, t]; >> Solve[b == 0, t] >> >> %%%%%%%%%%%%%%% >> >> Best regards, >> >> Gianluca Gorni > > I am not an expert on this sort of thing, but "mathematical common sense" > suggest to me that this may not be easy. Let's consider carefully the > problem of finding a "radical" expression for Cos[Pi/8] (or indeed any > Tr[Pi*m] where Tr is a trigonometric function and m a rational). Although > they do not look like it at first sight these expressions are in fact > algebraic numbers. The point is that Cos[Pi/8] is just the real part of the > cyclotomic number Cos[2*Pi/16]+I*Sin[2*Pi/16] which is just Root[#^16 - 1 &, > 16] as you can see from: > > In[54]:= > (Root[#^16 - 1 &, 16] - Cos[2*Pi/16] - I*Sin[2*Pi/16]) // FullSimplify > Out[54]= > 0 > > So we can write Cos[Pi/8] as 1/2(Root[#^16 - 1 &, 16] + 1/Root[#^16 - 1 &, > 16]), which is, of course, an algebraic number. Using RootReduce we can get > > In[58]:= > RootReduce[1/2(Root[#^16 - 1 &, 16] + 1/Root[#^16 - 1 &, 16])] > Out[58]= > 2 4 > Root[1 - 8 #1 + 8 #1 & , 4] > > and then > > In[60]:= > ToRadicals[%] // FullSimplify > Out[60]= > Sqrt[2 + Sqrt[2]] > ----------------- > 2 > > Of course in this case we could have much easier got this expression > starting from the known values of the trigonometric functions of Pi/4 and > then using half angle formulas. But my point is that there is a procedure > that can be tried in general in such cases. > > However, conversely: given a radical expresion e.g. v=((-1 + Sqrt[5]))/8 + > (Sqrt[(3*(5 + Sqrt[5]))/2])/4 how do you go about deciding if it is the real > or complex part of a cyclotomic number? In fact, v= Cos[Pi/15] but I > doubt that there is any algorithm which has a reasonable chcance of > determining if an arbitrary radical expression (with value between -1 and 1, > say) is the value of a trigonometric function of some rational multiple of > Pi. > (If I am wrong I would like to hear about this). > >