Chapter 6.2, Problem 61E

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°. If the distance from the earth to the moon is 240,000 mi, estimate the distance from the earth to the sun.

To determine

To find: The distance from the earth to the sun.

Answer to Problem 61E

The distance from the earth to the sun is $91.7\text{ million mi}$.

Explanation of Solution

Formula used:

1. The six trigonometric ratios are given by,

$\text{sin }\theta =\frac{\text{opposite}}{\text{hypotenuse}}$, $\text{cos }\theta =\frac{\text{adjacent}}{\text{hypotenuse}}$, $\text{tan }\theta =\frac{\text{opposite}}{\text{adjacent}}$.

$\text{csc }\theta =\frac{\text{hypotenuse}}{\text{opposite}}$, $\text{sec }\theta =\frac{\text{hypotenuse}}{\text{adjacent}}$, $\text{cot }\theta =\frac{\text{adjacent}}{\text{opposite}}$.

Calculation:

Given that the right angle forms when the moon is exactly half full, the earth, moon and sun.

Distance from the earth to the moon is $240,000\text{mi}$ and angle formed by the sun, earth and moon is $89.85°$.

Substitute $240,000$ for adjacent and $\theta =89.85°$ in $\text{cos }\theta =\frac{\text{adjacent}}{\text{hypotenuse}}$,

$\begin{array}{c}\cos 89.85°=\frac{240,000}{\text{hypotenuse}}\\ \text{hypotenuse}=\frac{240,000}{\cos 89.85°}\left[\because \cos 89.85°=0.00262\right]\\ =\frac{240,000}{0.00262}\\ =91,603,053.4\end{array}$

$\approx 91.7\text{million}$

Distance from the earth to the sun is 91.7 million.

Thus, the distance from the earth to the sun is $91.7\text{ million mi}$.