Troubleshooting Vector Angles in Three Dimensions

[whynot314]

ok so I am not sure how else to do this problem, my answers look similar to those in the book. where they have β=52.5 $\alpha=121$ $\gamma=53.1$, I feel like the book is wrong because i don't see any other way to solve this problem.

[URL=http://s1341.photobucket.com/user/nebula-314/media/20130523_174427_zpsda388c38.jpg.html][PLAIN]http://i1341.photobucket.com/albums/o745/nebula-314/20130523_174427_zpsda388c38.jpg[/URL][/PLAIN]

Heres how I solved for the angles
[URL=http://s1341.photobucket.com/user/nebula-314/media/20130523_174442_zps9be6b8d1.jpg.html][PLAIN]http://i1341.photobucket.com/albums/o745/nebula-314/20130523_174442_zps9be6b8d1.jpg[/URL][/PLAIN]

 

[whynot314]

The gammas are the same so I am unsure where I went wrong.