Decompose into Fourier series with respect to $\sin ((n+\frac{1}{2})x)$ ($n=0,1,\ldots$) on interval $[0,2\pi]$ and sketch the graph of the sum of such Fourier series:
a. $1$;
b. $x$;
c. $x(\pi -x)$;
d. $\sin (m x)$ with $m\in \mathbb{N}$;
e. $\cos (m x)$ with $m\in \mathbb{N}$;
f. $\sin ((m-\frac{1}{2}) x)$ with $m\in \mathbb{N}$.
