\text { Let } V \text { be the inner product space of continuous functions on }[-\pi, \pi] \text { with inner product } \langle f, g\rangle=\int_{-\pi}^{\pi} f(x) g(x) d(x)

Consider the function f E V given by: f(x)=2|\sin (x)|-1 . (a) Draw a sketch of this function. (b) Find coefficients an and b, such that f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos (n x)+b_{n} \sin (n x) for all x E [(-pi, pi]. (c) Consider the subspace W=\operatorname{span}\{\cos (x), \sin (x), \cos (2 x), \sin (2 x)\} of V. Find the projection f = Projw (f) of f(:, -).onto W with respect to the inner product