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This website concisely explains why the coefficient of friction is tan Î¸, where Î¸ is the angle between the ramp and horizontal
http://physics.bu.edu/~duffy/semester1/c6_measuremus.html

For static friction, Î¸ is the minimal angle that the allows an object to slip down.
For kinetic friction, 
θ is the minimal angle that allows the object to move at a constant non-zero speed.
In practice, it is typically easier to determine coefficient of static friction but if the ramp is long enough you can adjust the angle until some motion sensor tells you that the object is moving at a constant speed. 
In physics problems that you see in your textbook, almost always coefficient of kinetic friction is easier to find because you are often told that block is moving at constant speed.

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