N-th roots of complex numbers?
- Subject: N-th roots of complex numbers?
- From: Aleksas Domarkas
- Date: Wed, 2 Oct 2013 18:03:10 +0300
see:
http://www.math.utexas.edu/pipermail/maxima/2013/034149.html
Jaime Vilatte:
Sure, but the question is:
is there a simple process to realize that sin(atan(37/55)/3)=1/sqrt(26)
and cos(atan(37/55)/3)=5/sqrt(26)?
Can that process be implemented in Maxima?
Example. Compute sin(atan(37/55)/3), cos(atan(37/55)/3).
(%i1) r:x=sin(atan(37/55)/3);
(%o1) x=sin(atan(37/55)/3)
(%i2) asin(%);
(%o2) asin(x)=atan(37/55)/3
(%i3) %*3;
(%o3) 3*asin(x)=atan(37/55)
(%i4) sin(%);
(%o4) sin(3*asin(x))=37/(13*sqrt(26))
(%i5) trigexpand(%);
(%o5) 3*x*(1-x^2)-x^3=37/(13*sqrt(26))
(%i6) s:solve([%], [x]);
(%o6) [x=-(5*sqrt(78)+sqrt(26))/52,x=(5*sqrt(78)-sqrt(26))/52,x=1/sqrt(26)]
(%i7) sublist(s,lambda([e],is(abs(rhs(r-e))<10^-10)));
(%o7) [x=1/sqrt(26)]
(%i8) r:x=cos(atan(37/55)/3);
(%o8) x=cos(atan(37/55)/3)
(%i9) acos(%);
(%o9) acos(x)=atan(37/55)/3
(%i10) %*3;
(%o10) 3*acos(x)=atan(37/55)
(%i11) cos(%);
(%o11) cos(3*acos(x))=55/(13*sqrt(26))
(%i12) trigexpand(%);
(%o12) x^3-3*x*(1-x^2)=55/(13*sqrt(26))
(%i13) s:solve([%], [x]);
(%o13) [x=-(sqrt(78)+5*sqrt(26))/52,x=(sqrt(78)-5*sqrt(26))/52,x=5/sqrt(26)]
(%i14) sublist(s,lambda([e],is(abs(rhs(r-e))<10^-10)));
(%o14) [x=5/sqrt(26)]
best
Aleksas D