determine the static coefficient of friction if a box will be rasied at rest, right up until the point that the board on which it is sitting gets raised to 53 degrees.

Projection on x-axis:

0 = m•g•sinα – F(fr),
Projection on y-axis:
0 = - m•g•cos α +N.
Since F(fr) = k•N = k• m•g•cos α,
0 = m•g•sinα - k• m•g•cos α,
k = tanα