Okay so to do this in a way that's relevant for the MCAT, you should try and figure out which common trig angles Tan(x) = 5 would lie between. Think about what ratio the tangent represents. It's the Sin/Cos. So really when you see 'Tan(x) = 5', really what that means is "the Sin of x is five times greater than the Cos of x".
Now, Think about your unit circle:
At 45 degrees, the triangle made within the unit circle is one where the horizontal side and the vertical side are equal. Thus, the sin and cos are equal. Sin = Cos = sqrt2/2, so the ratio Sin/Cos = 1. So this is smaller than 5, so this angle is too small.
At 90 deg, the vertical side becomes maximal (1), and the horizontal side becomes minimal (0). Here Sin = 1, and Cos = 0. The limit of Sin/Cos = infinity. So 90 is too big an angle.
So where then, is the angle where "The sin is 5 times greater than the Cos". Well it would have to be above 45 degrees, and below 90 right?
Okay, that's a start, now think about 60 deg, At that point, what's the ratio between the sin and the cos? Well the vertical side is still larger than the horizontal side of the triangle, Sin = sqrt3/2 and Cos = 1/2. So Sin/Cos = sqrt3. Well Sqrt 3 is still smaller than 5, so the angle must be larger than 60.
Usually, on the MCAT, if you know that the angle is larger than 60 and smaller than 90, that would be enough to answer the question. There would probably only be one feasible answer. In this case, you could guess that the answer is about 70 or 80.
If you get comfortable with thinking about the concepts like this, you'll find that many problems that seem hard, are actually 'estimatable'. Remember not to get bogged down in solving everything explicitly. This is sometimes (mostly) not needed for the MCAT. Just get it down to a point where you can estimate the answer. GL.
