The problem with your answer arcsin(2/3), i.e. 0.730 radians to three places, is that an angle measuring at 0.730 radians (41.8 deg.) lies in the first quadrant. But, according to your post, u lies in quadrant II. To remedy this problem, we must find a 2nd quadrant angle with sine also equal to 2/3. It will likely serve you well to commit the following identity to memory. Sin(pi-x) = sin(x) [equivalently, sin(180-x)=sin(x) for degree measured angles] So, sin(u)=sin(pi-0.730). Clearly, pi-0.730, i.e., 2.41 radians [138 deg] lies in Q2. Thus, u=2.41 to two places.
Re. question 2, cos(3x)=0.5 indeed implies 3x=60 deg. But what other angle has cosine = 0.5? We have another identity stating that cos(t)=cos(2pi-t). Therefore, cos(60)=cos(360-60) or, cos(300). The result is: 3x=60, 3x=300 which implies x=20, x=100.
In the morning, I will supplement this post with more information that should clear up remaining confusion.