http://www.mapleprimes.com/posts/42356-Simultaneous-Sine-Functions en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 20:49:40 GMT Wed, 10 Jun 2026 20:49:40 GMT The latest comments added to the Post, simultaneous sine functions http://www.mapleprimes.com/images/mapleprimeswhite.jpg http://www.mapleprimes.com/posts/42356-Simultaneous-Sine-Functions http://www.mapleprimes.com/posts/42356-Simultaneous-Sine-Functions?ref=Feed:MaplePrimes:simultaneous sine functions:Comments#comment79242 It seems as if your system can be solved without tricks, <pre>t[1],t[2],t[4],t[6]:=0,delta,t[3]+delta,t[5]+delta: u1:=(t[2]-t[1])+(t[4]-t[3])+(t[6]-t[5])=U/2: delta:=solve(u1,delta): eq1:=expand(add((-1)^(i+1)*cos(omega*t[i]),i=1..6)=0): eq2:=expand(add((-1)^(i+1)*sin(omega*t[i]),i=1..6)=0): eq3:=expand(add((-1)^(i+1)*cos(r*omega*t[i]),i=1..6)=0): eq4:=expand(add((-1)^(i+1)*sin(r*omega*t[i]),i=1..6)=0): _EnvAllSolutions:=true: _EnvExplicit:=true: s:=solve({eq1,eq2,eq3,eq4},{t[3],t[5],r}); 12 Pi _Z1~ 2/3 Pi + 2 Pi _Z4~ s := {r = ----------, t[5] = ------------------, omega U omega -2/3 Pi + 2 Pi _Z6~ 12 Pi _Z1~ t[3] = -------------------}, {r = ----------, omega omega U -2/3 Pi + 2 Pi _Z4~ 2/3 Pi + 2 Pi _Z6~ t[5] = -------------------, t[3] = ------------------} omega omega </pre>Note, that both solutions satisfy condition <code> r*&omega;*U/12/Pi = an integer</code>, <pre>about(_Z1,_Z4,_Z6); Originally _Z1, renamed _Z1~: is assumed to be: integer Originally _Z4, renamed _Z4~: is assumed to be: integer Originally _Z6, renamed _Z6~: is assumed to be: integer</pre>Substituting integer values for _Zs, we obtain 2 series of solutions, for example, <pre>s1:=eval(s,[_Z1=1,_Z4=1,_Z6=1]); 12 Pi 8 Pi 4 Pi s1 := {r = -------, t[5] = -------, t[3] = -------}, omega U 3 omega 3 omega 12 Pi 4 Pi 8 Pi {r = -------, t[5] = -------, t[3] = -------} omega U 3 omega 3 omega eval([eq1,eq2,eq3,eq4],s1[1]); [0 = 0, 0 = 0, 0 = 0, 0 = 0]</pre> __________ Alec Mihailovs <a href="http://mihailovs.com/Alec/">http://mihailovs.com/Alec/</a> The latest comments added to the Post, simultaneous sine functions 79242 Sun, 29 Oct 2006 10:53:02 Z alec alec http://www.mapleprimes.com/posts/42356-Simultaneous-Sine-Functions?ref=Feed:MaplePrimes:simultaneous sine functions:Comments#comment79241 In SOLVE command, Why do you include the variable r? Is there any way to use the following command instead of your command? s1: = solve({eq1,eq2,eq3,eq4},{t[3],t[5]}); Could you explain why you use "r" variable which is positive real? Thank you in Smoking Section. Y-G Sung The latest comments added to the Post, simultaneous sine functions 79241 Sun, 29 Oct 2006 12:34:10 Z sungyg sungyg http://www.mapleprimes.com/posts/42356-Simultaneous-Sine-Functions?ref=Feed:MaplePrimes:simultaneous sine functions:Comments#comment86260 Well, one of r, U, or &omega; has to be added to t[3] and t[5] -because of the restriction that I mentioned - solutions exist only if <em> r&omega;U/(12&pi;)</em> is an integer. Otherwise, solve won't give an answer. _________ Alec Mihailovs <a href="http://mihailovs.com/Alec/">http://mihailovs.com/Alec/</a> The latest comments added to the Post, simultaneous sine functions 86260 Sun, 29 Oct 2006 12:43:19 Z alec alec http://www.mapleprimes.com/posts/42356-Simultaneous-Sine-Functions?ref=Feed:MaplePrimes:simultaneous sine functions:Comments#comment90906 Thanks a lot, Dr. Alec. However, I try to get the same answer as yours by setting up the equations. Here is my code. I think that s2 should be the same as s1. Please tell me why it does not give me the same results. Thank you Sung restart: t[1] := 0:t[2] := delta:t[4] := t[3]+delta:t[6] := t[5]+delta: u1 := t[2]-t[1]+t[4]-t[3]+t[6]-t[5] = 1/2*U: delta := solve(u1, delta): eq1 := expand(sum((-1)^(i+1)*cos(omega*t[i]), i = 1 .. 6) = 0): eq2 := expand(sum((-1)^(i+1)*sin(omega*t[i]), i = 1 .. 6) = 0): eq3 := expand(sum((-1)^(i+1)*cos(r*omega*t[i]), i = 1 .. 6) = 0): eq4 := expand(sum((-1)^(i+1)*sin(r*omega*t[i]), i = 1 .. 6) = 0): _EnvAllSolutions := true: _EnvExplicit := true: s1:=solve({eq1, eq2, eq3, eq4}, {r, t[3], t[5]}); trs1 := simplify(eq1*sin(omega*delta)+eq2*(1-cos(omega*delta))): trs2 := simplify(eq3*sin(r*omega*delta)+eq4*(1-cos(r*omega*delta))): rs1 := factor(trs1): rs2 := factor(trs2): s2:=solve({rs2, rs1}, {r, t[3], t[5]}); The latest comments added to the Post, simultaneous sine functions 90906 Wed, 01 Nov 2006 09:38:23 Z sungyg sungyg http://www.mapleprimes.com/posts/42356-Simultaneous-Sine-Functions?ref=Feed:MaplePrimes:simultaneous sine functions:Comments#comment90907 The system {rs1,rs2} in addition to solutions of the system {eq1,eq2,eq3,eq4} has other solutions. For example, when the lhs of the first term of trs1 equals minus lhs of the second term, and similarly for trs2, and these lhs' are not necessarily zeros, or even when they are zeros, it could be because of cos(&omega;*&delta;)=cos(r*&omega;*&delta;)=1 independently of values of eq1, eq2, eq3, and eq4. __________ Alec Mihailovs <a href="http://mihailovs.com/Alec/">http://mihailovs.com/Alec/</a> The latest comments added to the Post, simultaneous sine functions 90907 Wed, 01 Nov 2006 13:13:10 Z Alec Mihailovs Alec Mihailovs