Final answer:
By measuring the angle (x) and the distance (y) between the Earth and the Sun, and using trigonometric functions, a scientist can calculate the distance from the Moon to the Sun. Therefore, the correct answer is option a) Explain the relationship between the angle (x) and the distance (y).
Step-by-step explanation:
Understanding the Calculation of Distance from the Moon to the Sun
When the moon forms a right triangle with the Earth and the Sun during one of its phases, a scientist can use the measurements of the angle (x) and the distance (y) between the Earth and the Sun to calculate the distance between the moon and the Sun.
The angle (x) provides information on the relative positions of the Earth, Moon, and Sun. The distance (y) is the known distance between the Earth and the Sun. Given these measurements, and using trigonometric functions, the scientist can calculate the distance from the Moon to the Sun by performing the following steps:
Identify the lengths of the sides of the right triangle formed during this phase. The distance (y) is the hypotenuse of the right triangle.
Calculate the length of the side adjacent to the angle (x), which represents the distance from the Earth to the Moon, using the cosine function: Adjacent Side = y * cos(x).
With this Earth-Moon distance, apply the Pythagorean theorem or the sine function to find the distance from the Moon to the Sun: Opposite Side = y * sin(x).
The right triangle formed by the Earth, Moon, and Sun is significant because it allows for the application of basic geometric and trigonometric principles to calculate celestial distances.
The measurements taken by the scientist provide invaluable data to understand the celestial relationships and dynamics of our solar system, aiding in predictions of phenomena such as eclipses and phases of the moon.
Therefore, the correct answer is option a) Explain the relationship between the angle (x) and the distance (y).