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Since haversines have come up recently, it might be useful to some folks to know this simple relationship. The phase of the Moon (in the sense of percent illumination) is equal to the haversine of the angle from the Sun to the Moon, or in other words, the Moon-Sun lunar distance. As an equation, we could write it:

phase = hav(LD).

This is useful if you have any reason to calculate the phase of the Moon, but for us, it's probably more useful as a mnemonic of the variation of the haversine function with angle. When the Moon is new, the phase is zero, and of course, the haversine of 0° is zero. When the Moon is half full, the phase is 0.5 (or 50%) and at that time the LD is 90° and of course hav(90°)=0.5. And when the Moon is full (100% illuminated), the angle between the Sun and Moon approaches 180° and the haversine of 180° naturally is one. If you're trying to picture the variation of the haversine function in you head, think back to the variation of the phase of the Moon over a month. There's no genius here. It's just as easy to calculate (1-cos(x))/2, but this relationship with the phase of the Moon might be, as I'm suggesting, a quick aid to the memory. It has worked for me at least!

I sometimes bring up this relationship when discussing lunars. The phase of the Moon is a measure of the Sun-Moon lunar distance. In that sense, cultures worldwide have been keeping time by lunars for thousands of years. You can count days by observing the phase of the Moon. A sextant observation of an exact value of lunar distance, an exact measure of the angle LD, could be seen as a precise observation of the phase of the Moon.

-FER