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.

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

The angle θ means angle created on earth

i.e. angle BCA = θ = 89.85 degree

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

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

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