3. Consider a string on an instrument that is plucked by displacing a point at x =3L/4 by a distance A. One could proceed using brute force to find the

coefficients of the fourier series using the formulas I gave in lecture. A more elegant way is to notice that if you reflect the shape through the origin you end up with this. And this must be represented only by sines because cosines do not have the same reflection symmetry as the string displacement. So by developing the fourier series for the interval[-L, L], one can later restrict it to [0, L] without losing anything. (a) Find the value of I_{m n}=\int_{-L}^{L} d x \sin (\pi m x / L) \sin (\pi n x / L) (b) Use the result of (a) to find the coefficients Cm for the fourier representation of the string displacement y(x)=\sum_{i=1}^{\infty} c_{m} \sin (\pi n x / L) (c) Use a compute to plot the first seven terms.