The sinus approximate solution is based on the Taylor devepoped series of the sin function:

Recurrent expressions:
for n=0
T1=1;
T2=x;
T3=1;

for n=1
T1=-T1;

T2=x*x*T2;

T3=1; (for n=0)
T3=3! (for n=1)
for n=2
T3=5! =5*4*T3 (the previous one).

for n:
T3=(2n+1)!;

for n+1:
T3=(2(n+1)+1)!=(2n+2+1)!=(2n+1+2)!=(2n+3)!=(2n+3)*(2n+2)*(2n+1)!=
(2n+3)*(2n+2)*T3 (the previous one).