Question

Express sin^5 x
as a linear combination of sin kx and coskx, k belongs to Z.

Answer 1

$sin^5(x) = (\frac {e^{ix}\ -\ e^{-ix}} {2i} )^ 5 = \frac 1 {16\ *\ 2i}\ *\ (e^{ix}\ -\ e^{-ix})^5=\newline \frac 1 {16\ *\ 2i}\ *\ \newline (e ^ {5ix} - 5e^{3ix} + 10e^{ix} - 10e^{-ix} + 5e^{-3ix} - e^{-5ix}) = \newline = \frac 1 {16} (sin\ 5x\ -\ 5sin\ 3x\ +\ 10sin\ x) = \newline = \frac 1 {16}sin\ 5x\ -\ \frac 5 {16}sin\ 3x\ +\ \frac 5 {8}sin\ x$