Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

As you may have guessed, there is a simple formula for expressing great-circle distance directly in terms of longitude and latitude. Let \(\left(\alpha_{1}, \beta_{1}\right)\) and \(\left(\alpha_{2}, \beta_{2}\right)\) be the longitude- latitude coordinates of two points on the surface of the earth, where we interpret \(\mathrm{N}\) and \(\mathrm{E}\) as positive and \(\mathrm{S}\) and \(\mathrm{W}\) as negative. Show that the great-circle distance between these points is \(3960 \gamma\) miles, where \(0 \leq \gamma \leq \pi\) and $$ \cos \gamma=\cos \left(\alpha_{1}-\alpha_{2}\right) \cos \beta_{1} \cos \beta_{2}+\sin \beta_{1} \sin \beta_{2} $$

Step-by-Step Solution

Answer

The great-circle distance is 3960 miles times \( \gamma \), where \( \cos \gamma = \cos(\alpha_1-\alpha_2)\cos \beta_1 \cos \beta_2 + \sin \beta_1 \sin \beta_2 \).

Step 1: Recognize the Formula for Great-Circle Distance

The great-circle distance, the shortest path between two points on a sphere, can be given using the central angle \( \gamma \) between the two points. This angle is calculated using longitude and latitude values within the equation provided for \( \cos \gamma \). Thus, the distance is expressed as 3960 being the Earth's radius in miles (approx.).

Step 2: Understand the Given Cosine Formula

The given formula for \( \cos \gamma \) is \[ \cos \gamma = \cos(\alpha_1 - \alpha_2) \cos \beta_1 \cos \beta_2 + \sin \beta_1 \sin \beta_2 \]This uses trigonometric identities and represents the cosine of the angle \( \gamma \) between two vectors on a sphere, determined by their geographical coordinates.

Step 3: Apply the Law of Cosines to a Spherical Triangle

In spherical trigonometry, the formula for \( \cos \gamma \) parallels the law of cosines from planar trigonometry, adapted to spheres. It accommodates the angles (great-circle distance) between two longitude-latitude pairs.

Step 4: Derive the Angle \( \gamma \) from Great-Circle Distance

To relate \( \gamma \) to distance, recognize that the angle \( \gamma \) represents the arc on the circle in radians. Multiplying \( \gamma \) by the sphere's radius (3960 miles) gives you the great-circle distance: \[ 3960 \gamma \]

Key Concepts

Spherical Trigonometry

Spherical trigonometry is a branch of geometry that deals with the relationships between angles and sides on a sphere, unlike planar trigonometry which is concerned with flat surfaces. In spherical trigonometry, lines on the sphere are arcs of great circles, the shortest path between two points on a sphere. This is why the great-circle distance, which is the focus of this exercise, is of particular interest here.

Spherical triangles, formed by three great circles, have properties that are different from those of planar triangles. For example, the sum of the angles of a spherical triangle always exceeds 180 degrees. The Law of Cosines for spherical triangles, which we use when calculating the angle \(\gamma\), is slightly modified from its planar counterpart to accommodate the curvature of the sphere.

This concept is crucial when dealing with geographic coordinates because the Earth is not flat, and thus, distances along its surface must account for its round shape.

Longitude and Latitude

Longitude and latitude are the coordinate system used to determine any location on the Earth's surface. Each location is specified by a pair of numerical coordinates: longitude, which indicates position east or west, and latitude, which indicates position north or south. For calculations, these coordinates often need to be converted from degrees to radians.

The exercise uses these spherical coordinates to compute the great-circle distance — a measure of the shortest path between two points over the Earth's surface. In doing so, it highlights how distances on a sphere differ from those on a flat map, underscoring the importance of longitude and latitude in navigation and geography.

• Longitude ranges from 0° to 180° east and west from the prime meridian.
• Latitude ranges from 0° at the equator to 90° at the poles.

Understanding these coordinates and how they interact is foundational for applying spherical trigonometry concepts to real-world problems.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions and are essential tools in simplifying the calculations in spherical trigonometry.

In this exercise, the identity of the cosine of the difference of two angles is used, which states:
\[ \cos(\alpha_1 - \alpha_2) = \cos \alpha_1 \cos \alpha_2 + \sin \alpha_1 \sin \alpha_2 \]

This identity helps compute the central angle \(\gamma\) between two locations based on their latitude and longitude. It reflects how these relationships can be leveraged to simplify otherwise complex problems involving angular separations on a sphere.

Additionally, understanding and applying the identity:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)
• \( \sin (a \pm b) = \sin a \cos b \pm \cos a \sin b \)

can further aid in solving various geometric problems, including the calculation of angles and distances on the Earth's surface.