How to find the angle between straight lines, one of which intersects the center of the Moon and the Earth, and the other the center of the Sun and the Earth?

Question

Mathematics

"A scientist named Aristarchus tried to explain the structure of the universe using the geocentric system. However, as he continued to observe, he had one doubt: he noticed that the waxing and waning of the moon occurs due to a change in the angle of incidence of light from the Sun. This means that when you see half of the moon, the Sun should illuminate it exactly from the side.
To find the relative position of the Sun and the Moon (and the ratio of the sizes of the Earth, the Moon and the Sun), it is enough to determine the angle (which is asked in the title of the question).
Aristarchus determined that the value of the angle = 87 degrees (in fact, 89.85).
How did Aristarchus find this value (in 310-230 BC)? And how was it determined more accurately over time?

Answer 1

E - 3D coordinates of the center of the Earth
M - 3D coordinates of the center of the Moon
S - 3D coordinates of the center of the Sun
dotproduct(A, B) - dot product of vectors A and B
arc(A) - inverse cosine of angle A
angle = arc( dotproduct( ME, SE) ) ;

Answer 2

There is one guess ...
He extended one hand to the Moon, and the other to the Sun. And he measured the angle between his hands. So that's what I determined.