MaplePrimes - comments on Blog Entry, Converting Half-Angle Trig Formulas to Radicals

MaplePrimes - comments on Blog Entry, Converting Half-Angle Trig Formulas to Radicals http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Thu, 11 Jun 2026 16:24:13 GMT Thu, 11 Jun 2026 16:24:13 GMT The latest comments added to the Blog Entry, Converting Half-Angle Trig Formulas to Radicals http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Blog Entry, Converting Half-Angle Trig Formulas to Radicals http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals simplify,size http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals?ref=Feed:MaplePrimes:Converting Half-Angle Trig Formulas to Radicals:Comments#comment101758 The <a href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=simplify/size">simplify(...,size)</a> command is sometimes useful. But not always... it might not suffice for handling the 3rd column of table 2. <pre>> x:=1/6^(1/2)*(3-sqrt(5))^(1/2); (1/2) 1 (1/2) / (1/2)\ - 6 \3 - 5 / 6 > simplify(radnormal(x),size); 1 (1/2) / (1/2) \ - 3 \5 - 1/ 6 > x:=1/6^(1/2)*(3+sqrt(5))^(1/2); (1/2) 1 (1/2) / (1/2)\ - 6 \3 + 5 / 6 > simplify(radnormal(x),size); 1 (1/2) / (1/2) \ - 3 \5 + 1/ 6 </pre> The latest comments added to the Blog Entry, Converting Half-Angle Trig Formulas to Radicals 101758 Wed, 16 Feb 2011 21:32:15 Z pagan pagan factor http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals?ref=Feed:MaplePrimes:Converting Half-Angle Trig Formulas to Radicals:Comments#comment101826 factor does it (except for pulling the common factor out): <pre>factor(1/6^(1/2)*(3-sqrt(5))^(1/2)); 1/2 1/2 1/2 3 5 3 - ---- + --------- 6 6 factor(1/6^(1/2)*(3+sqrt(5))^(1/2)); 1/2 1/2 1/2 3 5 3 ---- + --------- 6 6 </pre> Tracing shows that it does it by writing (3-5^(1/2))^(1/2)=-a/2+a*b/2, where a^2=2 and b^2=5, i.e. basically the same idea: <pre>infolevel[all]:=5: trace(factor): factor(1/6^(1/2)*(3-sqrt(5))^(1/2)); [...] radnormal/rational/basis: basis is [5, 3, 2] [...] 1/2 1/2 (3 - 5 ) = -1/2 %1 + 1/2 %2 %1], [...] 2 %1 := RootOf(_Z - 2, index = 1) 2 %2 := RootOf(_Z - 5, index = 1) [...] </pre> Then, using factor and applyrule, similarly to the calculation by hand, a simple and uniform approach to all these nine cases is possible, with quite compact results in most of them: <pre>r1:=sin(y::algebraic/2)=sqrt((1-cos(y))/2): r2:=cos(y::algebraic/2)=sqrt((1+cos(y))/2): r3:=tan(y::algebraic/2)=csc(y)-cot(y): (factor@applyrule)([r1,r2,r3],sin(arcsin(2/3)/2)); 1/2 1/2 3 5 - ---- + ---- 6 2 (factor@applyrule)([r1,r2,r3],cos(arcsin(2/3)/2)); 1/2 1/2 3 5 ---- + ---- 6 2 (factor@applyrule)([r1,r2,r3],tan(arcsin(2/3)/2)); 1/2 5 3/2 - ---- 2 (factor@applyrule)([r1,r2,r3],sin(arccos(2/3)/2)); 1/2 1/2 2 3 --------- 6 (factor@applyrule)([r1,r2,r3],cos(arccos(2/3)/2)); 1/2 1/2 10 3 ---------- 6 (factor@applyrule)([r1,r2,r3],tan(arccos(2/3)/2)); 1/2 5 ---- 5 (factor@applyrule)([r1,r2,r3],sin(arctan(2/3)/2)); / 1/2\1/2 | 6 13 | |2 - -------| \ 13 / ---------------- 2 (factor@applyrule)([r1,r2,r3],cos(arctan(2/3)/2)); / 1/2\1/2 | 6 13 | |2 + -------| \ 13 / ---------------- 2 (factor@applyrule)([r1,r2,r3],tan(arctan(2/3)/2)); 1/2 13 ----- - 3/2 2 </pre> The latest comments added to the Blog Entry, Converting Half-Angle Trig Formulas to Radicals 101826 Sat, 19 Feb 2011 07:38:01 Z Alejandro Jakubi Alejandro Jakubi applyrule is elegant, but... http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals?ref=Feed:MaplePrimes:Converting Half-Angle Trig Formulas to Radicals:Comments#comment101924  I am edified by the comment demonstrating the use of rules and the applyrule command. However, the rules for sine and cosine are quadrant-dependent. This approach is useful when the quadrant of the angle is known. Otherwise, a sign error could be introduced. But the illustration of the use of applyrule is quite valuable and has been added to the Little Red Book.  RJL Maplesoft The latest comments added to the Blog Entry, Converting Half-Angle Trig Formulas to Radicals 101924 Wed, 23 Feb 2011 19:37:39 Z rlopez rlopez conditional http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals?ref=Feed:MaplePrimes:Converting Half-Angle Trig Formulas to Radicals:Comments#comment101935 <a href="http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals#comment101924">@rlopez</a>  The quadrant-dependence can be handled by conditional rules, like: <pre>r:=[conditional(sin(x::algebraic) =sqrt((1-cos(2*x))*(1/2)), _is(sin(x)>=0)), conditional(sin(x::algebraic) =-sqrt((1-cos(2*x))*(1/2)), _is(sin(x)<0))]: applyrule(r,sin(u)) assuming 0<u,u<Pi; 1/2 1/2 (2 - 2 cos(2 u)) applyrule(r,sin(u)) assuming Pi<u,u<2*Pi; 1/2 -1/2 (2 - 2 cos(2 u)) </pre> Indeed, there are here two weaknesses in Maple that need to be addressed: 1. Properties like the expression x in [2*k*Pi, (2*k+1)*Pi] when k is integer. A good representation of predicates and "abstract" sets in their terms is missing. This is why I am using instead a condition on the sign of sin(x) for the case of x a name. 2. Handling properties, like the sign, with nested functions: <pre>is(sin((1/2)*arctan(v, u))>0) assuming v<0,u>0; FAIL </pre> Meeting weakness 2 is a consequence of weakness 1. I have made some progress in a workaround for 2. The latest comments added to the Blog Entry, Converting Half-Angle Trig Formulas to Radicals 101935 Wed, 23 Feb 2011 23:48:03 Z Alejandro Jakubi Alejandro Jakubi conditional http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals?ref=Feed:MaplePrimes:Converting Half-Angle Trig Formulas to Radicals:Comments#comment101941  Thanks, Alejandro. I appreciate your continued development of this topic.  RJL Maplesoft The latest comments added to the Blog Entry, Converting Half-Angle Trig Formulas to Radicals 101941 Thu, 24 Feb 2011 01:22:09 Z rlopez rlopez identify http://www.mapleprimes.com/maplesoftblog/101756-Converting-HalfAngle-Trig-Formulas-To-Radicals?ref=Feed:MaplePrimes:Converting Half-Angle Trig Formulas to Radicals:Comments#comment120376 <pre>Digits:=15: Matrix(3,(i,j)->identify(evalf([sin,cos,tan][i]([arcsin,arccos,arctan][j](2/3)/2)))); </pre> <a href="/view.aspx?sf=120376/383535/Matrix3.png"><img style="display: block; margin-left: auto; margin-right: auto;" src="/view.aspx?sf=120376/383535/Matrix3.png" alt="Matrix(3,3,{(1, 1) = 1/6*15^(1/2)-1/6*3^(1/2), (1, 2) = 1/6*6^(1/2), (1, 3) = 1/26*(338-78*13^(1/2))^(1/2), (2, 1) = 1/6*15^(1/2)+1/6*3^(1/2), (2, 2) = 1/6*30^(1/2), (2, 3) = 1/26*(338+78*13^(1/2))^(1/2), (3, 1) = 3/2-1/2*5^(1/2), (3, 2) = 1/5*5^(1/2), (3, 3) = -3/2+1/2*13^(1/2)},datatype = anything,storage = rectangular,order = Fortran_order,shape = [])"></a> The first 2 rows can be also obtained in a uniform fashion as <pre>Matrix(2,3,(i,j)->sqrt(combine([sin,cos][i]([arcsin,arccos,arctan][j](2/3)/2)^2))); </pre> And the 3rd row can be calculated as <pre>radnormal~(%[1]/~%[2],rationalized);</pre> _______________ Alec Mihailovs, PhD The latest comments added to the Blog Entry, Converting Half-Angle Trig Formulas to Radicals 120376 Wed, 25 May 2011 12:45:20 Z Alec Mihailovs Alec Mihailovs