Calculus of a Single Variable, Chapter 9, 9.1, Section 9.1, Problem 59

Sine function is periodic function with period of 2pi.
This means that the given sequence will have 12 unique values (because 12cdot pi/6=2pi) and these values will repeat cyclically, more precisely a_n=a_(n+12), forall n in NN. Therefore, we conclude that the given sequence is not monotonic.
On the other hand, codomain of the sine function is [-1,1] so the sequence is obviously bounded.
Maximum terms of the sequence are a_(3+12k)=1, k in ZZ, while the minimum terms are a_(9+12k)=-1, k in ZZ.
The image below shows the first 60 terms of the sequence. We can clearly see the periodic nature of the sequence.