In order to forecast the changes in the moon's appearance from day to day, we can rely on evidence from various sources such as the "Phases of the Moon" article, simulations, and moon sphere models. By examining these sources, we can gain a better understanding of the moon's orbit around the Earth, the angle at which the sun's light hits the moon's surface, and the resulting changes in the moon's illumination. This information allows us to predict the different phases of the moon, such as new moon, first quarter, full moon, and third quarter, and how they will appear from our vantage point on Earth.