The Waxing Crescent is the phase following the new moon, when a thin sliver of the Moon's lit face becomes visible and grows from night to night, its illumination increasing from a trace toward half. It appears low in the western sky after sunset and sets soon after. The crescent is the emblem of the new month in the lunar calendars, and its first sighting fixes the start of the Islamic month. Traditionally the waxing crescent is read as a time of emergence and gathering momentum, the first visible growth of what the new moon began. These are the astronomical facts of the phase together with the meanings traditions attach to it; the crescent is simply the sunlit edge of the Moon as it moves along its orbit away from the line between Earth and Sun.
The lunar cycle from one new moon to the next, called the synodic month, averages about twenty-nine and a half days, noticeably longer than the sidereal month of roughly twenty-seven and a third days it takes the Moon to return to the same position against the background stars; the difference exists because the Earth itself has moved along its own orbit around the Sun in the meantime, so the Moon must travel a little further to catch up and realign with the Sun as seen from Earth. The waxing crescent falls early in that longer synodic reckoning, and because the phases are what nearly every lunar and lunisolar calendar actually tracks, this is the phase, rather than the sidereal position, that determines the length of a calendar month in traditions from the Islamic Hijri year to the Hebrew and Chinese lunisolar systems.
Sources AstronomyAndrew Fraknoi, David Morrison, and Sidney C. Wolff (OpenStax) with The Oxford Companion to the Year: An Exploration of Calendar Customs and Time-ReckoningBonnie Blackburn and Leofranc Holford-Strevens