Reduce: This system cannot be solved with the methods available to Reduce

Question

I got the error message 'Reduce::nsmet: This system cannot be solved with the methods available to Reduce.'

Here is my code

neweqns = {Sin[a] + Sin[a - b] + Sin[a - c] == 0 && 
 Sin[a - b] == Sin[b] + Sin[b - c] && Sin[a] + Sin[c] == Sin[b] && 
 Sin[a - c] + Sin[b - c] == Sin[c]}

result = Reduce[neweqns && 0 <= a <= 2 π && 0 <= b <= 2 π && 
   0 <= c <= 2 π, {a, b, c}, Reals]

Could anyone help? Thank you!

Answer 1

Let us expand sines of differences, then Reduce can crack it.

newesteqns =  Sin[a] + Sin[a]*Cos[b] - Cos[a]*Sin[b] + Sin[a]*Cos[c] - 
Cos[a]*Sin[c] == 0 &&  Sin[a]*Cos[b] - Cos[a]*Sin[b] == 
Sin[b] + Sin[b]*Cos[c] - Cos[a]*Sin[c] &&  Sin[a] + Sin[c] == Sin[b] &&  
Sin[a]*Cos[c] - Cos[a]*Sin[c] + Sin[b]*Cos[c] - Cos[b]*Sin[c] == 
Sin[c];
result = Reduce[newesteqns && 0 <= a <= 2 \[Pi] && 0 <= b <= 2 \[Pi] && 
0 <= c <= 2 \[Pi], {a, b, c}, Reals]

a == \[Pi] && ((b == 0 && (c == 0 || c == \[Pi] || c == 2 \[Pi])) || (b == \[Pi] && (c == 0 || c == \[Pi] || c == 2 \[Pi])) || (b == 2 \[Pi] && (c == 0 || c == \[Pi] || c == 2 \[Pi])))) || (b == 0 && ((a == 0 && (c == 0 || c == \[Pi] || c == 2 \[Pi])) || (a == 2 \[Pi] && (c == 0 || c == \[Pi] || c == 2 \[Pi])))) || (b == \[Pi] && ((a == 0 && (c == 0 || c == \[Pi] || c == 2 \[Pi])) || (a == 2 \[Pi] && (c == 0 || c == \[Pi] || c == 2 \[Pi])))) || (b == 2 \[Pi] && ((a == 0 && (c == 0 || c == \[Pi] || c == 2 \[Pi])) || (a == 2 \[Pi] && (c == 0 || c == \[Pi] || c == 2 \[Pi]))))