Distance from Venus to the Sun The elongation αof a planet is the angle formed by the planet, ea

step 1 So for this question, we are given the sun, Venus, and Earth. And we are going to show the right

Distance from Venus to the Sun The elongation αof a planet...

Distance from Venus to the Sun The elongation $\alpha$ of a planet is the angle formed by the planet, earth, and sun (see the figure). When Venus achieves its maximum elongation of $46.3^{\circ},$ the earth, Venus, and the sun form a triangle with a right angle at Venus. Find the distance between Venus and the sun in astronomical units (AU). (By definition the distance between the earth and the sun is $1 \mathrm{AU}$.)

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Key Concepts

Astronomical Units

An astronomical unit (AU) is a standard unit of length used in astronomy to express distances within our solar system, defined as the average distance between the Earth and the Sun. This concept is fundamental when comparing the relative positions of planets.

Planetary Elongation

Planetary elongation refers to the angular separation between a planet and the Sun as observed from Earth. For inferior planets like Venus, the maximum elongation occurs when the planet’s position relative to Earth and the Sun forms a right triangle, facilitating the use of trigonometry to determine its distance from the Sun.

Right Triangle Geometry

This concept involves the properties of triangles that have one angle equal to 90°. In problems like this one, the right angle allows the use of trigonometric functions to relate the sides and angles, making it easier to solve for unknown distances.

Trigonometric Ratios

Trigonometric ratios, such as the cosine function, relate an angle of a right triangle to the ratio of two specific sides. In this context, the cosine of an angle is used to determine the length of one side when the hypotenuse is known.

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