Perhaps a better word than "precisely" is "concisely" - those angles require only a single term to have their trig functions expressed exactly. And if I remember correctly there are 2*pi radians in a rotation, not one.

Degrees, radians, and rotations are all measures of angle; I just feel the degree is the least graceful. Radians are extremely useful in mathematics and physics, as certain trig identities and other formulae only work when expressed in radians; in this manner radians are a sort of "natural" unit for angle. Rotations are also useful, but in different contexts, primarily engineering, where the important thing is not where on a circle one is located but how many times one has gone around it.

Degrees, however, are not as convenient or systematic, and the archour is intended to replace them with something easier to use for the times when we want part of a circle expressed as a rational number. One objective with this unit is to make the bridge between basic angle units (degree or archour) and radians easier for students. This shift is easier for archours because one archour (15 degrees) is pi/12 radians, two archours (30 degrees) is pi/6 radians, three archours (45 degrees) is pi/4 radians, four archours (60 degrees) is pi/3 radians, and six archours (90 degrees) is pi/2 radians - simple fractions of pi in radians, small integers in archours, and concise trig functions all around. The archour is the greatest common divisor of these common angles.