!
!   ******   function T_sine   *****************************************
!
     function T_sine(x,n) 

      implicit none 
      integer:: k, n
      real(8):: pi
      real(8):: x, xp, fact_k, ak, fk
      real(8):: T_sine_k
      real(8):: T_sine
      real(8),parameter:: eps = 1.0d-10

      pi = 4.0d0 * atan(1.0d0)
!
!   initalize: f(x=0) = sin(0), 0! = 1, x**0 = 1.0
!
      T_sine_k = 0.0d0
      fact_k = 1.0d0
      xp = 1.0d0
!
!   take summation sigma_( a(i)*x**i ) for i = 1 to n
!
      do k = 1,n
        fact_k = fact_k * dfloat(k)  ! calculate factorial real(k!)
        ak = (-mod(k,2))**(k/2) / fact_k  ! coefficient

        xp = xp * x     ! calculate xp = x**k
        fk = ak * xp    ! k-th order term

        T_sine_k = T_sine_k + fk  ! calculate summation
      end do
      
      T_sine = T_sine_k

      return
      end function T_sine