Question

When the moon is exactly half full, the earth, moon, and sun form a right triangle. The right angle is the angle at the moon. At that time, the angle formed by the sun earth and moon is 89.85◦ (the angle at the Earth). If the distance from the earth to the moon is 240,000 miles, estimate the distance from the earth to the sun.

Answer 1

The distance from earth to sun is 91,673,351.94 miles

Solution:

Given that the earth, moon, and sun form a right triangle shown in figure

The figure is attached below

Point A represents moon

Point B represents sun

Point C represents earth

The right angle is the angle at the moon. This is represented by point A in right angle triangle

Given that angle at earth is 89.85 degree

Angle BCA = 89.85 degree

Distance between earth and moon = AC = 240,000 miles

Since, Right angled triangle is formed we can use trigonometric identities

`\cos \theta=\frac{B a s e}{\text {Hypotenuse}}`

The angle θ means angle created on earth

i.e. angle BCA = θ = 89.85 degree

`\cos \left(89.85^(\circ)\right)=(A C)/(B C)`

Let BC = d, which is the distance between Earth and the sun

`\mathrm{D}=(240,000)/(\cos (89.85))=91,673,351.94 \mathrm{miles}`

So the distance from earth to sun is 91,673,351.94 miles