Thanks for another math question - my favorite.
Slope expressed as a percent means that in 100 feet of horizontal
distance, the (road, trail, whatever) has fallen by n%.
A little trigonometry will get you degrees. We know that the tangent
of an angle is the opposite side divided by the adjacent side. The
arctangent then gives the angle when you give it the opposite /
adjacent fraction. Draw your triangle like this (I hope the ASCII art
A | \
That looks to me to be about a 100% grade (100 feet out, 100 feet
down). Label the two sides of the triangle as A and B, and the angle
we are considering is C. arctan(A/B) = C. Fix B to be 100 feet, and
then A is the percent grade.
If you have a 10% grade, then you compute the angle C as
arctan(10/100) = 5.7106 degrees. If you have a 100% grade, then we
see the formula works: arctan(100/100) = 45. If you want the other
angle, you use a bit of geometry. First off, we have fixed thet
bottom left angle to be 90 degrees, and we know the sum of the angles
in a triangle is 180. Therefore, the top angle is 90 - C degrees (180
degrees total - 90 degrees in bottom left angle - C. I simplified.)
FYI: arctan is also called the inverse tangent, and is written as
tan^-1 (with the exponent formatted better!).