Question 11.4.2

The ancient Greek astronomer and mathematician Aristarchus wanted to know how far away the Sun and Moon were. He reasoned that when the phase of the Moon was exactly half full, the Sun, Moon, and Earth would form a right triangle. He measured the angle between the Sun and half-full Moon in the sky to be $87^\circ$. Diagram: aristarchus 1

  1. Using this measurement, how many times farther away is the Sun than the Moon? If you don’t have a calculator, use the small angle approximation $\sin\theta\approx \theta$ (where $\theta$ is measured in radians). Show answer

  2. Aristarchus’ measurement was not very accurate. The real angle is about $89^\circ51’$ (it varies a little over the years). Use this to find the true ratio between the distance of the Sun and Moon. Show answer