During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase ...

Question

During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be 89.85 degrees, the ratio of the Earth-Sun and Earth-Moon distances is closest to

A. 328   B. 382   C. 238   C. 253

Accepted Answer

ES/EM = 1/cos(89.85 deg) approx 382 At half-moon, the Sun-Moon-Earth angle is exactly 90 degrees because the Sun illuminates exactly half the Moon's face as seen from Earth. This means Earth, Moon, and Sun form a right-angled triangle with the right angle at the Moon. Knowing the angle at Earth (Moon-Earth-Sun = 89.85 deg), we can use trigonometry to find the ratio of the two distances. The ratio ES/EM equals 1/cos(angle at Earth).