Chapter 6.2, Problem 65E

To determine

To find: The distance between Venus and the sun in astronomical unit.

Answer to Problem 65E

The distance between Venus and the sun is $0.723\text{AU}$.

Explanation of Solution

Given:

Given triangle when Venus achieves its maximum elongation, the earth, Venus and the sun form a triangle with a right angle at Venus.

Figure (1)

Sides of the triangle, When Venus achieves maximum elongation of $\alpha =46.3°$ and the distance between the earth and sun is $1\text{AU}$.

Calculation:

Use sin property of trigonometric ratio when Venus achieves maximum elongation to form a right triangle,

$\sin \alpha =\frac{\text{opposite}}{\text{hypotenuse}}$

Substitute $1\text{AU}$ for hypotenuse and $\alpha =46.3°$ in the above formula.

$\begin{array}{c}\sin 46.3°=\frac{\text{opposite}}{1}\\ \text{opposite}=1\times \sin 46.3\left\{\sin 46.3=0.723\right\}\\ =0.723\times 1\\ =0.723\text{AU}\end{array}$

Thus, the distance between Venus and the sun in astronomical units is $0.723\text{AU}$.