Compute the Fourier series of the function $f(x) = 3\sin\!\left(4x\right)$ on $[-\pi,\pi]$ using the trig identities

$\displaystyle \sin(u)\sin(v) = \frac{1}{2}\left[\cos(u-v)-\cos(u+v)\right]$

$\displaystyle \sin(u)\cos(v) = \frac{1}{2}\left[\sin(u+v)+\sin(u-v)\right]$

$a_0$

$=$

$\displaystyle\int$

$dx =$

$a_n$

$=$

$\displaystyle\int$

$dx$

$\displaystyle \mbox{Apply trig identity}$

$\hskip 10pt$

$=\frac{3}{2\pi}\Bigg[$

$\displaystyle\int$

$dx +$

$\displaystyle\int$

$dx \Bigg]$

$\displaystyle\hskip 15pt =$

$b_n$

$=$

$\displaystyle\int$

$dx$

$\displaystyle \mbox{Apply trig identity}$

$\hskip 10pt$

$=\frac{3}{2\pi}\Bigg[$

$\displaystyle\int$

$dx -$

$\displaystyle\int$

$dx \Bigg]$

$\hskip 10pt$

$=$

$n=4$ $n\neq4$