Use the spherical law of cosines (previous exercise) to...

Name: Problem 35, Chapter 9: A History of Mathematics: An Introduction
Uploaded: 2020-10-29T17:52:10-08:00
Duration: 1 min 56 s
Channel: Anurag Kumar
Description: Use the spherical law of cosines (previous exercise) to determine the great circle distance between New York and London (whose coordinates are given in Exercise 33 ). Again, assume that the circumference of the earth is 25,000 miles.

Use the spherical law of cosines (previous exercise) to determine the great circle distance between New York and London (whose coordinates are given in Exercise 33 ). Again, assume that the circumference of the earth is 25,000 miles.

Key Concepts

Conversion of Linear Distance to Angular Measure

When calculating distances on a sphere, it is often necessary to convert linear measurements into angular distances. This is done by using the relationship between the circumference of the sphere and its radius. Once the radius is known (derived from the given circumference), a linear distance can be expressed as a corresponding central angle, which is then used in spherical trigonometric formulas.

Spherical Coordinates

Spherical coordinates are used to define the positions of points on the surface of a sphere through a pair of angles, typically latitude and longitude. This coordinate system is fundamental in applying spherical trigonometry to practical problems, such as computing distances between geographical locations on Earth.

Spherical Law of Cosines

The spherical law of cosines is a fundamental formula in spherical trigonometry, linking the sides and angles of a spherical triangle. It allows one to calculate the angular distance between two points on a sphere using the cosine of the arc length, the cosines of the spherical coordinates, and the cosine of the angle difference. This concept is essential in determining distances on curved surfaces, such as the Earth.

Great Circle Distance

The great circle distance is the shortest distance between two points on the surface of a sphere. It is measured along the arc of a great circle, which is any circle that shares the same center and radius as the sphere. Understanding how to compute the great circle distance is crucial in fields such as navigation and geodesy, where accurate routing along the Earth's surface is required.

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