Spherical trigonometry

Spherical trigonometry is a special kind of math that helps us understand shapes and angles on a round ball, like the Earth. Instead of flat triangles, it deals with triangles drawn on the surface of a sphere, where the sides are curves called great circles. These great circles are the longest possible circles you can draw on a sphere, like lines of longitude or the equator on our planet.

This type of trigonometry is very important for many fields, especially astronomy, where it helps scientists figure out the positions of stars and planets. It is also essential in geodesy, the science of measuring the Earth, and in navigation, helping people find their way accurately over long distances.

The ideas behind spherical trigonometry began a long time ago in ancient Greece and were later expanded by mathematicians in Islamic countries. In more recent history, famous thinkers like John Napier made big advances, and by the end of the 1800s, the subject was fully developed thanks to books like Isaac Todhunter’s textbook. Today, new methods using vectors, quaternions, and computer calculations continue to improve how we use spherical trigonometry.

Preliminaries

A spherical polygon is a shape on the surface of a sphere, with its sides being arcs of great circles—the spherical version of straight lines in flat geometry. These polygons can have any number of sides more than one. For example, a two-sided spherical polygon is called a lune or digon, like the curved part of an orange slice. Three arcs make a spherical triangle, which is the main focus of this topic. Polygons with more sides can be thought of as being made up of several spherical triangles.

In spherical trigonometry, the points where the arcs meet are called vertices and are labeled with capital letters A, B, and C. The arcs between these points are called sides and are labeled with lowercase letters a, b, and c. The angles at each vertex can be measured in radians. Each triangle also has a matching shape called a polar triangle, which helps in solving problems by switching the roles of angles and sides.

Cosine rules and sine rules

Cosine rules

Main article: Spherical law of cosines

The cosine rule is a key idea in spherical trigonometry. It helps us understand the relationships between the sides and angles of triangles drawn on a sphere. This rule is like the cosine rule you might know from regular geometry, but it works for shapes on a round ball instead of a flat surface.

Sine rules

Main article: Spherical law of sines

The sine rule for spherical triangles tells us how the sizes of the angles and the lengths of the sides are connected. It’s similar to the sine rule in flat geometry but adjusted for the curved surface of a sphere.

Identities

Spherical trigonometry helps us understand the relationships between the sides and angles of triangles drawn on a sphere. Unlike flat triangles, these triangles are made up of arcs of great circles — the longest possible circles you can draw on a sphere.

This type of trigonometry is very important for tasks like predicting where stars will appear in the night sky, planning routes for ships and airplanes, and measuring the shape of the Earth. It uses special math rules to connect the pieces of these spherical triangles, allowing scientists and explorers to solve complex problems involving curved surfaces.

Solution of triangles

Main article: Solution of triangles § Solving spherical triangles

Spherical trigonometry helps us find missing pieces of a triangle on a sphere when we know some of its parts. For example, if we know three sides of a triangle, we can find the angles. If we know two sides and the angle between them, we can find the other pieces too. There are many different situations, like knowing three angles or two angles and a side, and each has its own way to solve it.

One common way is to split the triangle into two right-angled triangles and use special rules to find the missing parts step by step. This makes it easier to avoid mistakes when the angles get very small or very large. There are many methods to solve these triangles, and mathematicians have studied them for a long time.

Area and spherical excess

See also: Solid angle and Geodesic polygon

When you draw shapes on a sphere, like triangles, there’s something special called the spherical excess. This is how much bigger the angles of the shape add up to compared to what they would be on a flat surface. For a triangle on a sphere, if you add up its three angles and subtract 180 degrees (or π radians), what’s left is the spherical excess.

This idea helps us find the area of shapes on a sphere. For example, a triangle with three right angles (90 degrees each) has an excess of 90 degrees, showing how curved the sphere is! These ideas are important for navigation, mapping, and understanding the night sky.

Main article: Girard's theorem