T0 = 1;
t = 0:0.02:T0;
w0 = 2*pi/T0;

plot(t, cos(w0*t))
   hold on
  plot(t, sin(3*w0*t),'r')
   hold off

% Finding the expansion for  f(t) = t 

k = 1;

dt = 0.01;
t = 0:dt:T0; 

ak = 2*dt*sum( cos(w0*k*t).* t);

bk = 2*dt*sum( sin(w0*k*t).* t);

approx1 = 1/2 +  ak * cos(w0*k*t) +  bk * sin(w0*k*t);

plot(t,t);
  hold on 
plot(t, approx1,'.');