Just to be sure GunNSpot is clear.
• The phenomenon you saw was NOT the earth’s shadow be cast from the sun onto the moon.
• If what you saw was a crescent moon “east” of the setting sun, then the moon was in fairly close proximity to the sun. The moon you saw was waxing toward Full, which means that when the moon is Full, it will be exactly opposite the sun (i.e. the sun will be setting in the west, just as the moon will be rising in the east).
• That explains WHY you saw a crescent. Imagine yourself in the center of a room (you’re the earth), and you’re holding a baseball (the moon). The west wall of the room is mounted a strong light, which simulates the setting sun. Now, hold the baseball out at arm’s length, and position it exactly between you and the light. You’re going to see only the ball’s silhouette. THIS is the New Moon. It’s completely in shadow. What makes the shadow? Not the earth, of course, it’s the “daylight” side of the baseball, that is in the way of the “nightside”, so it casts a shadow on itself. At any given time, the baseball (or moon) will be 50% illuminated.
Now, start slowly rotating yourself (keeping the baseball out at arms length) counter-clockwise. You’ll notice the baseball “waxing” a crescent shape. This arises because you’re starting to position the ball in a way where you can see the “daylight” side creeping in. That is, you’re moving the ball out from between yourself and the light. Keep turning CCW. The ball will continue rolling through these “phases” until the ball appears completely illuminated, this is the Full Moon. You’ll notice you’re the one that stands in between the ball and the light, as the ball is now pointed at the East wall. If your shadow is on the ball, that’s a lunar eclipse. But, in space, the moon, earth, and sun are relatively small, and far way from each other, so it’s not all that common for them to align so accurately. So typically the moon misses the earths shadow.
You can keep turning around until the baseball goes through all it’s phases (after Full it begins to wan). It’ll return back to a crescent right before it goes back to New Moon again, then the whole process repeats itself.
So, in this way, I’m able to completely understand where the moon is relative to the earth, just by looking at it’s phase. From now on, anytime you see a Full Moon, you’ll immediately notice that it’s rising just as soon as the sun is setting. Pretty cool.