Problem 4. (a) Suppose :(0,)R admits a Fourier sine series (x)=n=1bnsin(nx). Find the Fourier sine series for the function (x)=(x)cosx. Particularly, if admits a Fourier sine series (x)=(x)cosx=n=1b~nsin(nx), express b~k 's in terms of bk 's. (Hint: 2sin(A)cosB=sin(A+B)+sin(AB). Reindex the summation.) (b) The Fourier sine series of the function (x)=x on (,) is given by x=2k=1n1sin(nx). Find the solution u of the wave equation with Dirichlet condition and wave speed c=1 : {utt=uxxu(t,0)=u(t,)=0(t,x)(0,)(0,),t>0, with initial data , : u(0,x)=(x)=(x)cos(x),ut(0,x)=(x)=sin(x). Express the solution u in terms of Fourier (sine or cosine) series. Problem 4. (a) Suppose :(0,)R admits a Fourier sine series (x)=n=1bnsin(nx). Find the Fourier sine series for the function (x)=(x)cosx. Particularly, if admits a Fourier sine series (x)=(x)cosx=n=1b~nsin(nx), express b~k 's in terms of bk 's. (Hint: 2sin(A)cosB=sin(A+B)+sin(AB). Reindex the summation.) (b) The Fourier sine series of the function (x)=x on (,) is given by x=2k=1n1sin(nx). Find the solution u of the wave equation with Dirichlet condition and wave speed c=1 : {utt=uxxu(t,0)=u(t,)=0(t,x)(0,)(0,),t>0, with initial data , : u(0,x)=(x)=(x)cos(x),ut(0,x)=(x)=sin(x). Express the solution u in terms of Fourier (sine or cosine) series