NavList:

Lu Abel, you wrote:
"Just remember 1,2,3,3,2,1  (which sums to 12). My understanding is that the Rule of Twelfths is accurate to 2 or 3% of the true sine wave"

This is a very old rule from an era when integers were essential to teaching and comprehension. Today, wouldn't it be just as easy today to remember the values that are accurate to three significant digits: 0.80, 2.20, 3, 3, 2.20, 0.80? That's the change in the value of the sine for every 30° change in the angle, expressed as sixths. Better yet, why not just memorize the sine values for the angles in question, especially since there are only four independent values: sin(0°)=0, sin(30°)=1/2=0.5, sin(60°)=sqrt(3)/2=0.8660..., sin(90°)=1. We can throw in one more that's easy to remember: sin(45°)=sqrt(2)/2=0.7071... This short list completely covers the "Rule of Twelfths" as a special case, and it's 100% accurate.

Note: the little rule I gave in my first post here is not contained in the above; it's independent. But you can get close using the "Rule of Twelfths" (or its exact, mocern equivalent as detailed above). The sine values alone tell you that, in one half period, from maximum to minimum, the value of the quantity changes by 2 units (from +1 to -1). They also tell you that in the middle third of that range, the sine value change by 1 unit (from +0.5 to -0.5) which means that the average rate of change in the middle third is 1.5 times the average rate of change in the whole range from max to min. As I noted yesterday, the actual maximum rate (in the pure sine curve case) is pi/2 or 1.57 times the average rate. So it's just a question of the level of detail you seek... what sorts of numbers you're comfortable memorizing... and how much time you want to spend working it out.

Frank Reed