If the initial conditions at t=0 are given by x(0)=Xo and V(0)=Vo, the
position X of a simple harmonic oscillator at later time t is
expressed by:
X(0)=Xocoswt+(Vo/w)sinwt, where w is the angular velocity.  We will
consider a similar but slightly different situation, where the initial
conditions are now given at a finite time t=t1 byt X(t1)=X1 and
V(t1)=V1.
We assume the general solution in the form X(t)=Acoswt+Bsinwt.  Using
the initial conditions at t1, write down the simultaneous equations
that A and B must satisfy.

if X(t)=Xo*cos(wt)+(Vo/w)*sin(wt), then you can differentiate to get 
V(t) = dX/dt = -Xowsinwt+Vo*cos(wt).

if X(t) = Acos(wt)+Bsin(wt), V(t)=-wAsin(wt)+wBcost(wt), so if we're
given X(t1)=X1 and V(t1)=V1, we know that

X1=Acos(w*t1)+Bsin(w*t1) and V1=-wAsin(w*t1)+wBcos(w*t1)

if you know w and t1, it's obviously easy to eliminate the sin's and
cos's and solve for A and B, or, if t1 is a multiple of Pi/2, again
it's easy to simplify.

there's probably a way to write these formulas with A= and B= on the
left, but I can't think of a way to do it that's possible to type into
a text box, so I'll hope the form I gave was what you wanted.  If not,
I hope I've at least pointed you in the right direction!

if something isn't making sense, please do ask for a clarification....

thanks,
David

Be sure to cite this properly when you turn that homework problem in.