SOLUTION: Please help me to prove using de Moivre's Theorem that {{{ 4*sin^3 (x) =3*sin(x) -sin(3*x)}}}

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Please help me to prove using de Moivre's Theorem that {{{ 4*sin^3 (x) =3*sin(x) -sin(3*x)}}}      Log On


   

Question 1034445: Please help me to prove using de Moivre's Theorem that +4%2Asin%5E3+%28x%29+=3%2Asin%28x%29+-sin%283%2Ax%29
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
,after applying de Moivre's and direct expansion.
= cos%5E3%28x%29%2B3icos%5E2%28x%29sinx+-+3cosx%2Asin%5E2%28x%29+-+isin%5E3%28x%29.
Now equate the imaginary parts of the first complex quantity and the last complex quantity.
==>
==> sin%283x%29+=+3sinx+-+3sin%5E3%28x%29+-+sin%5E3%28x%29+=+3sinx+-+4sin%5E3%28x%29
==> sin%283x%29+=+3sinx+-+4sin%5E3%28x%29, or highlight%284sin%5E3%28x%29+=+3sinx+-+sin%283x%29%29.
As an added bonus, what would happen if you equated the real parts of the first complex quantity and the last complex quantity? Try it out yourself.