Verify graphically the odd-even identity for the model in #5 by substituting for and

Chapter 5, Problem 473

(choose chapter or problem)

Modeling the Illumination of the Moon

From the Earth, the Moon appears to be a circular disk in the sky that is illuminated to varying degrees by direct sunlight. During each lunar orbit the Moon varies from a status of being a new Moon with no visible illumination to that of a full Moon, which is fully illuminated by direct sunlight. The U.S. Naval Observatory has developed a mathematical model to find the fraction of the Moon’s visible disk that is illuminated by the Sun. The data in the table below (obtained from the U.S. Naval Observatory Web site, http://aa.usno.navy.mil/, Astronomical Applications Department) show the fraction of the Moon illuminated at midnight for each day in January 2010.

Verify graphically the odd-even identity \(\sin (-\theta)=-\sin (\theta)\) for the model in #5 by substituting \(-\theta\) for \(\theta\) and graphing  \(-a \sin (-\theta)+k\). How does this model compare to the original one?