Problem 61 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

The distance from the earth to the sun is approximately 93,819,432 miles.

Step 1: Understand the Geometry

The problem involves a right triangle where the earth, moon, and sun form a right angle. The earth is the vertex of this right angle, with the moon and sun at the other two vertices respectively.

Step 2: Identify Given Information and Unknown

We are given that the angle between the sun, earth, and moon is \(89.85^{\circ}\), and the distance between the earth and the moon is 240,000 miles. We need to find the distance between the earth and the sun, which we'll denote as \(d\).

Step 3: Apply Trigonometric Ratios

Using the right triangle, apply the tangent function: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 89.85^{\circ}\), the opposite side is the distance from the earth to the moon (240,000 miles), and the adjacent side is the unknown distance \(d\).

Step 4: Solve for the Earth-Sun Distance

Rearrange the tangent formula \(\tan(89.85^{\circ}) = \frac{240,000}{d}\) to solve for \(d\). We calculate \(d = \frac{240,000}{\tan(89.85^{\circ})}\). Since \(\tan(89.85^{\circ})\) is a very large number, \(d\) will be significantly larger than 240,000.

Step 5: Calculate the Distance

Using a calculator, evaluate \(\tan(89.85^{\circ}) \approx 382.77\). Substitute in the formula: \(d = \frac{240,000}{382.77} \approx 93,819,432 \text{ miles}\).

Key Concepts

Right Triangle Geometry

In the context of this problem, right triangle geometry is crucial. The Earth, moon, and sun form a right triangle with the Earth at the right angle. Understanding a right triangle is fundamental, as it involves one angle being exactly 90 degrees. The side opposite the right angle is known as the hypotenuse, the longest side in a right triangle. Here, the Earth forms the right angle while the moon and the sun represent the other two vertices. Recognizing this configuration allows us to use trigonometric functions effectively to find unknown distances, using the known distance from Earth to the moon as one of the triangle's sides.

Tangent Function

The tangent function is a key trigonometric function used to solve this exercise. For any angle in a right triangle, the tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, it is expressed as:

• tan(θ) = \(\frac{\text{opposite}}{\text{adjacent}}\)

In this problem, the opposite side is the distance from the Earth to the moon, which is 240,000 miles.
The adjacent side, which we are trying to determine, is the distance from the Earth to the sun.
By understanding the tangent function, we can rearrange this equation to find the missing side length.

Angle Measurement

Angle measurement plays a pivotal role in using trigonometry to calculate distances. In this problem, the angle between the sun, Earth, and moon is measured at approximately \(89.85^{\circ}\) degrees.
When dealing with such an angle, it is almost 90 degrees but slightly less. This small degree difference results in a large tangent value because tan(90°) is theoretically undefined, showcasing the tangent function's sensitivity to angle changes near 90 degrees.
Precision in angle measurement is critical, especially in astronomical calculations, as even tiny errors can result in huge discrepancies over large distances.

Distance Estimation

Estimating the distance between Earth and the sun using given data requires careful calculation. Since the angle is very large but not exactly 90 degrees, small changes in the angle result in significant differences in the tangent value.
To solve for the distance to the sun, we use the equation:

• \(d = \frac{240,000}{\tan(89.85^{\circ})}\)

This formula allows us to solve for \(d\), the Earth-sun distance, based on the triangle's configuration.
Computational tools enable us to evaluate the tangent accurately, resulting in a calculated distance of approximately 93,819,432 miles.
This process highlights how trigonometry can translate angular measurements into distance estimations across unimaginably vast spaces.