I actually already have the solution to this, but would just like some clarification of how the solution was reached.

The solutions provided used the fact that by Taylor's theorem, 

$\sin x = T\_6(x) + R\_6(x) $

The rest of the solution I understand, but how do you know to use the 6th degree Taylor polynomial? Why not 5th or 7th?

Any explanation would be greatly appreciated! Thanks so much!