Lunar Sidereal and Synodic Periods
The Moon’s sidereal orbital period (the sidereal month) is ~27.3 days; this is the time interval that the Moon takes to orbit 360° around the Earth relative to the “fixed” stars.
The period of the lunar phases (the synodic month), e.g. the full moon to full moon period, is longer at ~29.5 days.
At New Moon, the Moon is aligned with the Sun. During the lunar month the Earth orbits (revolves) ~30° around the Sun and the Moon orbits (revolves) ~390° to align with the Sun again.
If PSID is the Moon’s sidereal orbital period (in days), then the Moon revolves around the Earth at a rate of 360 / PSID ° per day. So in a time interval t days, the Moon revolves around the Earth t × 360 / PSID °.
If PYEAR is the Earth’s sidereal orbital period (in days), then the Earth revolves around the Sun at a rate of 360 / PYEAR ° per day and so in t days the Earth revolves around the Sun t × 360 / PYEAR °.We wish to determine the time interval when the angle that the Moon has revolved around the Earth minus 360° is equal to the angle the Earth has revolved around the Sun.
Hence t × 360 / PSID − 360 = t × 360 / PYEAR
Divide by t × 360, so
1 / PSID − 1 / t = 1 / PYEAR
By definition t = PSYN .
Now PSID = 27.32158 days, PYEAR = 365.25636 days,
hence PSYN = 29.53049 days.