Half Moon

[CentaurZ]

We see a waxing Half Moon in the evening sky when the Moon has completed the First Quarter of a Lunation.
We see a waning Half Moon in the morning sky when the Moon has begun the Last Quarter of a Lunation. 

🙂

[CentaurZ]

The Moon is always in one of the four quarters of a lunation, and what's listed as FQ on calendars is the cusp marking the completion of the first quarter and the commencement of the second quarter. As Jean Meeus notes, a more proper term is Eastern Quadrature, i.e. 90 degrees east of the Sun geocentrically. An average of 17 minutes before that time the Moon displays geocentric dichotomy, i.e. a 50% illuminated Half Moon. Beware, some authors get these confused.