Tim, the realtionship between the tangent of an angle and the angle itself is non-linear.
Angles are ratios; the measure of an angle in radians is the arc length/radius. In other words we are measuring the arc lengths of parts of a circle of unit radius. The tangent of the angle is the rise/run.
Since we are not measuring the same quantities to begin with, halves or other fractions of the original won't be equal either.
Trigonometric Functions of a Unit Circle
