Lunar cycle and the orbital period
A lunar cycle takes 29.5 days, while a full rotation of the Moon around Earth takes 27.3 days. Common sense would state that these two numbers should be the same. They are not because the Moon has to catch up to its starting position, which has changed. This change was a result of the Earth rotating around the sun.
You may have personally observed that it takes the moon one month to go through a Moon phase cycle. It is not quite a month from new Moon to new Moon, but rather 29.5305882 days. This is called a synodic orbital period, and is the time it takes as we observe from Earth for the Moon to move back to the same position.
If you were to view the same Moon cycle from some point inside our solar system above Earth, then the time it would take for a full cycle would be 27.3217 days, roughly two days less. This is called a sidereal orbital period.
So why are these two different? Well, from Earth, we are viewing the Moon from a moving platform. We are on Earth and Earth is moving. This alters our angle of view with respect to the Moon, and as a result alters the phase. The Earth's orbital direction is such that it lengthens the period for earthbound observers.
Although the synodic and sidereal periods are exact numbers, the Moon phase can't be precisely calculated by simple division of days because the motion of the Moon is affected and perturbed by various forces of different strengths. In this case, the Moon’s motion is referred to as its orbital speed and position. Hence, complex equations are used to determine the exact position and phase of the Moon at any given point in time. As a simple rule when determining rotational speed however, it can be assumed that the Moon rotates at 10 miles per hour, compared to the Earth, which rotates at about 1000 miles per hour.
As a final note, the lunar cycle is incredibly predictable. However, on a scale beyond our lifetime, it is changing very slowly. Every year, the Moon moves an addition 3.8 centimeters further away. This is happening because the Moon is “stealing” some of Earth’s rotational energy, using it to propel itself further and further from Earth’s gravitational well. It is hypothesized that shortly after its formation, the Moon was only 14,000 miles away from Earth. Currently, the Moon is 280,000 miles away. Eventually, the Moon will escape from Earth’s gravitational field entirely.