Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the great-circle distance from St. Paul (longitude \(93.1^{\circ} \mathrm{W}\), latitude \(45^{\circ} \mathrm{N}\) ) to Turin, Italy (longitude \(7.4^{\circ} \mathrm{E}\), latitude \(\left.45^{\circ} \mathrm{N}\right)\).

Step-by-Step Solution

Answer

The great-circle distance between St. Paul and Turin is approximately 7,089 kilometers.

Step 1: Convert Longitudes to Radians

First, convert the longitudes of St. Paul and Turin from degrees to radians. Use the formula: \[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]For St. Paul: \[93.1^{\circ} \times \frac{\pi}{180} = 1.624 \, \text{radians} \text{ (West is negative)} = -1.624 \, \text{radians}\]For Turin: \[7.4^{\circ} \times \frac{\pi}{180} = 0.129 \, \text{radians}\]This gives the longitudes as \(-1.624\) and \(0.129\) radians, respectively.

Step 2: Apply the Great-Circle Distance Formula

The great-circle distance can be found using the formula:\[\Delta \sigma = \cos^{-1} \left( \sin \phi_1 \sin \phi_2 + \cos \phi_1 \cos \phi_2 \cos(\lambda_2 - \lambda_1) \right)\]where \(\phi_1 = \phi_2 = 45^{\circ} \text{ (latitude in radians,} \phi = 45^{\circ} \times \frac{\pi}{180} = 0.785 \, \text{radians)}\). Substitute the values into the formula:\[\Delta \sigma = \cos^{-1} \left( \sin(0.785) \sin(0.785) + \cos(0.785) \cos(0.785) \cos(0.129 - (-1.624)) \right)\]Simplifying inside the cosine inverse gives the distance angle \(\Delta \sigma\).

Step 3: Calculate the Angular Separation

Calculate:\[\Delta \sigma = \cos^{-1} \left( \sin^2(0.785) + \cos^2(0.785) \cos(1.753) \right)\]First, find \(\sin(0.785) = \cos(0.785) = \frac{\sqrt{2}}{2}\), and then:\[= \cos^{-1} \left( \frac{1}{2} + \frac{1}{2} \cos(1.753) \right)\]Calculate \(\cos(1.753)\) and substitute it back in to find \(\Delta \sigma\).

Step 4: Convert Angular Separation to Distance

Find the Earth’s radius and compute the great-circle distance using:\[D = R \Delta \sigma\]where the Earth's radius \(R\) is approximately 6,371 kilometers. After finding \(\Delta \sigma\) in radians, multiply by \(6,371\) km to convert the angular distance into linear distance.

Key Concepts

Latitude and Longitude Conversion

To solve the problem of finding the great-circle distance, it's essential to understand how to convert the geographical coordinates of latitude and longitude from degrees to radians.
When working with trigonometric functions, radians are the preferred unit because they allow for direct application in these functions.

To convert degrees to radians, use the formula:

• Radians = Degrees × \( \frac{\pi}{180} \)

For example, St. Paul’s longitude is \(93.1^{\circ} \text{W}\).
Converting this into radians involves multiplying by \( \frac{\pi}{180} \), resulting in approximately \(-1.624 \text{ radians} \).
The westward longitude is typically negative in mathematical calculations.
For Turin, Italy, the longitude is \(7.4^{\circ} \text{E}\), which converts to about \(0.129 \text{ radians} \).

Remembering these steps will ensure you accurately perform conversions needed for any problem involving great-circle distances on the Earth, enhancing not only this specific solution but also your broader understanding of geographical calculations.

Trigonometric Functions

Trigonometric functions play a pivotal role in calculating great-circle distances because they relate to angles and circular measurements.
For great-circle distances, we utilize the cosine inverse function, or \(\cos^{-1}\), within the spherical distance formula.

Why are trigonometric functions necessary here? These functions help us understand the relationships between the angles formed by latitudes, longitudes, and the Earth's radius.
In the exercise, we calculate the angle \(\Delta \sigma\), which is key for identifying the arc length between St. Paul and Turin using

• \(\Delta \sigma = \cos^{-1} \left( \sin \phi_1 \sin \phi_2 + \cos \phi_1 \cos \phi_2 \cos(\lambda_2 - \lambda_1) \right)\)

Here, \(\sin\) and \(\cos\) refer to the trigonometric sine and cosine functions applied to the latitudinal and longitudinal measurements in radians.
Knowing how to effectively employ these functions is crucial as they bridge the gap between geographic and mathematical calculations, facilitating accurate distance measurements on a spherical surface like Earth.

Angular Separation Calculation

Angular separation calculation is essential when determining the shortest path between two points on a globe, known as the great-circle distance.
This process involves finding \(\Delta \sigma\) through angular relationships.

To calculate angular separation, use the formula:

• \(\Delta \sigma = \cos^{-1} \left( \sin^2(0.785) + \cos^2(0.785) \cos(1.753) \right)\)

In this setup, \(\phi\) values are pre-converted latitudinal angles, and we must also deal with the longitudinal angle difference \(\lambda_2 - \lambda_1\), as seen in the steps.
Given \(\phi_1 = \phi_2\), the computation simplifies slightly but still requires understanding the precise alignment of Earth's coordinates.
Calculate \(\sin(0.785)\) and \(\cos(0.785)\) as \(\frac{\sqrt{2}}{2}\) for simplification.

This solution ultimately provides the angle in radians that, when multiplied by Earth's radius, gives the linear distance between the points.
Mastering this step offers invaluable insight into how abstract trigonometric formulas translate to tangible geographical measurements, allowing for accurate calculations of real-world distances.