Original Message (ID '276572') By Bill Simpson: 
I cannot tell whether you are using fractional minutes for angles or minutes and seconds for angles, so this shows both method.
Watch out for using characters like that little x, which you think means multiply and Mathematica thinks otherwise. You can use * or sometimes just use a space between two terms.
Probably unlike your calculator, Mathematica cautions you that there are multiple solutions and shows you two of them. You can eliminate this warning if you provide additional information about the domain of z that you are interested in.
In[1]:= Solve[Cos[z] == (Sin[(33 + 13/60 + 6/36000)°]  Sin[(23 + 55/60 + 4/3600)°]*Cos[(109 + 10/60 + 9/3600)°])/(Cos[(23 + 55/60 + 4/3600)°]*Sin[(109 + 10/60 + 9/3600)°]), z]// N
From In[1]:= Solve::ifun: Inverse functions are being used by Solve so some solutions may not be found; use Reduce for complete solution information.
Out[1]= {{z> 2.47937},{z>2.47937}}
In[2]:= Solve[Cos[z] == (Sin[(33 + 13.6/60)°]  Sin[(23 + 55.4/60)°]*Cos[(109 + 10.9/60)°])/(Cos[(23 + 55.4/60)°]*Sin[(109 + 10.9/60)°]), z] // N
From In[2]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
Out[2]={{z> 2.48},{z>2.48}} 
