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.

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iwillanswer iwillanswer · Answer 1 · 2019-12-04T04:52:49+01:00

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

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

[tex]\cos \theta=\frac{B a s e}{\text {Hypotenuse}}[/tex]

The angle θ means angle created on earth

i.e. angle BCA = θ = 89.85 degree

[tex]\cos \left(89.85^{\circ}\right)=\frac{A C}{B C}[/tex]

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

[tex]\mathrm{D}=\frac{240,000}{\cos (89.85)}=91,673,351.94 \mathrm{miles}[/tex]

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