The distance from the Sun to Earth is...

The distance from the Sun to Earth is approximately149,600,000 km. The distance from the Sun to Venus is approximately $108,200,000 \mathrm{~km}$. The elongation angle $\alpha$ is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Venus. Suppose that the elongation angle for Venus is $10^{\circ} .$ Use this information to find the possible distances between Earth and Venus.

Key Concepts

Astronomical Distance Measurement

Measuring astronomical distances involves using various methods and the observed quantities like angles and relative distances in the solar system. These measurements rely on a combination of geometric principles and observational data to accurately determine vast expanses such as the distances between Earth, the Sun, and other planets.

Law of Cosines

The Law of Cosines is a trigonometric tool that relates the lengths of sides in any triangle to the cosine of one of its angles. In astronomical contexts, it is frequently used when two sides of a triangle (for example, the distances from the Sun to each planet) and the included angle (the elongation angle) are known, allowing the calculation of the third side (the distance between the two planets).

Elongation Angle

The elongation angle is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to another planet. It is an important concept in observational astronomy, as it helps determine the relative positions of the planets with respect to the Earth–Sun line, which in turn can be used to calculate distances and understand the apparent motion of planets in the sky.

Triangle Geometry

Triangle geometry, particularly when applied to the celestial triangle made by Earth, the Sun, and another planet, is crucial in solving problems involving distances between bodies in the solar system. By relating sides and angles of this triangle, one can utilize mathematical relationships to determine unknown distances between these objects.

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