Why did Aristarchus make his measurements of the Sun’s distance at the time of a half Moon?

Aristarchus made his measurements about the distance of sun in the time of half moon. There is a great and very deep insight of his for this reasoning. We know that when you are looking at the half moon while the sun is still there in the sky, then the sun light must be incidenting on earth and moon at right angles. That’s an extremely good intuition made by Aristarchus. By this he got to know that the sun rays, moon and earth makes a right angled triangle, whose 3 vertices are sun, moon and earth. According to the basic laws of trigonometry, if you know at least one angle except the right angle and at least the length of one side, then you can predict the length of the other two sides. The equations are, sin = perpendicular / hypotenuse cos = base / hypotenuse tan = perpendicular / base So, the only thing he had to measure was the angle between his line of sight and the moon as the distance between earth and moon he already knew. By using the above trigonometric relations, he was able to measure the distance between earth and sun with very high precision. This shows a work of a genius.