Knowing the angle between the directions from the Earth to the Moon and to the Sun, derive a formula that establishes the fraction of the Moon’s

Knowing the angle between the directions from the Earth to the Moon and to the Sun, derive a formula that establishes the fraction of the Moon’s diameter perpendicular to the line connecting its horns and measures the width of the illuminated part of the Moon. Note: Directions to the Sun from the Earth and from the Moon are considered parallel (they differ by less than 9 ‘).

If the Moon L is visible in the form of a crescent, then the angle a at the Earth T between the directions on it and on the Sun S is less than 90 °. Having built the drawing, we see that the desired value is x = 1 / 2- 1/2 cos a = sin ^ 2 * f / 2. , if the diameter of the moon is taken as 1.