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 looks at 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.

This type of trigonometry is very important for many fields, especially astronomy, where it helps scientists find 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 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. Its sides are arcs of great circles—the spherical version of straight lines. These polygons can have more than one side. For example, a two-sided shape 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 here. 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.

Cosine rules and sine rules

Cosine rules

Main article: Spherical law of cosines

The cosine rule is an important idea in spherical trigonometry. It helps us learn about the sides and angles of triangles on a sphere. This rule is like the cosine rule from regular geometry, but it works for shapes on a round ball, not a flat surface.

Sine rules

Main article: Spherical law of sines

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

Identities

Spherical trigonometry helps us understand triangles drawn on a sphere. Unlike flat triangles, these triangles are made up of arcs of great circles — the biggest circles you can draw on a sphere.

This type of trigonometry is 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, helping scientists and explorers solve problems with curved surfaces.

Solution of triangles

Main article: Solution of triangles § Solving spherical triangles

Spherical trigonometry helps us find missing parts of a triangle on a sphere when we know some of its pieces. 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 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, they act in a special way. This is called the spherical excess. It shows how much bigger the angles of the shape are compared to a flat surface.

For a triangle on a sphere, add up its three angles and subtract 180 degrees. What’s left is the spherical excess.

This 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. This shows how curved the sphere is! These ideas are useful for navigation, mapping, and studying the night sky.

Main article: Girard's theorem