I am quite familiar with Newton's laws and those introductory mechanics problems and exercises, but I am not going to check whether your reasoning is physically correct.

Either way, this is the correct syntax:

Solve[m*g*Sin[\[Beta]] == \[Mu]s1*m*g*Cos[\[Beta]], \[Beta] ]

I removed the Degree Symbol, and a few [ and ].

Note that you can't solve for a multipication of two variables in Solve, i.e., you are going to get the message

Solve::ivar: [Degree] [Beta] is not a valid variable.

So if you ask me, solve for the angle beta using Solve and get its value in radians, and convert it to degrees:

Mod[ \[Beta]/Degree,360]

And note that \[Beta] should be inserted as a numerical value.

Good luck!