Obviously particle will move to the right when velocity is >0. cos x>0 implys that x is in 1st or 4th quadrant. So for your function that is for 0+k*2pi<=2t<pi/2+k*2pi and

3pi/2+k*2pi<2t<=2pi+k*2pi,

where k is integer.

Deviding first inequality by 2 you get

0 + k*pi<=t<pi/4 + k*pi

and deviding second inequality by 2 you get

3pi/4 + k*pi <t<=pi + k*pi

So for k=0 and k=1 you get your solution.

Particle is moving ot the right when

t is from [0,pi/4] U [3pi/4,5pi/4] U [7pi/4,2pi].

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