Distance to the Sun When the moon is exactly half full, the...

Distance to the Sun When the moon is exactly half full, the earth, moon, and sun form a right angle (see the figure). At that time the angle formed by the sun, earth, and moon is measured to be $89.85^{\circ} .$ If the distance from the earth to the moon is $240,000 \mathrm{mi}$, estimate the distance from the earth to the sun.

Key Concepts

Right Triangle Geometry

This concept involves understanding the properties of triangles that have one 90° angle. In astronomical contexts, forming a right triangle with known sides and angles allows one to apply trigonometric relationships to calculate unknown distances. The right triangle setup simplifies the computation because the relationships between the sides and the angles are well-defined.

Trigonometric Ratios

Trigonometric ratios such as sine, cosine, and tangent are key tools in relating the angles of a triangle to the lengths of its sides. In problems involving celestial objects, these ratios help convert angular measurements into linear distance estimates, making it possible to deduce vast distances from relatively modest observed angles.

Small Angle Approximation

The small angle approximation is a mathematical simplification used when angles involved are very close to zero (or when the deviation from a reference angle is small). Although in this problem the angle is very close to 90° rather than 0°, the principle of approximating trigonometric functions for small deviations can be applied to ease the calculations, especially when determining the slight difference from the ideal right angle.

Astronomical Distance Measurement

This concept encompasses the methods used to estimate the distances between celestial bodies. By observing the geometric relationships and angular separations between objects (such as the Earth, Moon, and Sun), astronomers can use principles of triangulation and parallax to scale up from known distances to much larger, otherwise inaccessible, distances in space.

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