Model the phases of the moon with a trigonometric function

Question

A.) What would be a good function to model the phases of the moon?

B.) What determines the phase shift?

C.) What are the amplitude and midline?

Answer 1

1. A good function would be the cosine like Y = A cos (wt + Φ ), where A is max brightness (full moon) and we is angular frequency, t is time and phi is a phase angle. The product of wt is the number of degrees that the moon rotates through in a given time t, which is about 30 days for a full period of 360 degrees.
2. The phase shift is determined by phi, which is the angle that we view the moon from. At full the angle is zero. At new it is 180. At quarter it is 90. Note that the sun always illuminates one half of the moon, but its phase is due to its changing angular position as it orbits the earth.
3. As stated above the amplitude is A. The midline is A/2, as this make y = 0 at the new moon phase.