The sun iluminates half of the moon's surface, except during a lunar eclipse. The illuminated portion of the moon visible from Earth varies as t revolves around Earth resulting in the phases of the moon. The period from a full moon to a new moon and back o a full moon is called a synodic month and is 29 days, 12 hours, and 44.05 minutes long. Write a sine function that models the fraction of the moon's surface which is seen to be illuminated during a synodic month as a function of the number of days, d, after a full moon. (Note: full moon equals 1/2 illuminated.) SHOW WORK!

Can anyone help me fast.

NOTE: EDITED TO ADD THE +C TERM (forgot it in my original post)

A basic sine function is of the form:



Where f is the "frequency" of the sine wave, in this case in terms of cycles per day. It is related to the period (T) of the sine wave by:



So here you need to first find the period T in days (you are given T in terms of days hours and minutes, so you need to convert that to days).

The phase shift is a term that corrects for the fact that at t = 0, sin(ft) = 0, but you want f(0) to be at a maximum at t=0, so you need to shift the sine wave by 1/4 period. So needs to be set at 1/4 of the overall period (in days).

A is the half-amplitude of the sine wave - in this case you are told that the maximum illumination is considered to be 1/2, so you want a sine wave that varoes from 0 to 1/2. The half amplitude would be 1/4. (Personally I would have called it 1, not 1/2, since for an observer on the earth the full moon appears 100% illuminated. Your math teacher is apparently not much of an astronomer, but if s/he wants you to call a full moon 1/2 illuminated, so be it).

Finally, C needs to be set so that the sine wave function varies from 0 to 1/2. Without this term the sine wave would vary from -1/4 to +1/4. So make C equal to A.

So now can you put it all together?