5. To infinity …

‘To infinity … ’ looks at how infinite trigonometric series are used to compute π. It shows how Machin’s formula used the inverse tangent series to compute π to a hundred places. Lord Kelvin’s use of Fourier analysis in studying tide behaviour is also explained along with the Gibbs phenomenon. The invention of Cartesian coordinates and calculus in the 17th and 18th centuries reflects a major shift in mathematics. Geometry gradually changed from being synthetic (in the style of Euclid) to being analytic. The basic mathematical objects―originally points, lines, and shapes as well as numbers―became functions that accept input quantities and produce output quantities.

2. Sines, cosines, and their relatives

‘Sines, cosines, and their relatives’ begins by defining the basic trigonometric functions—sine, cosine, and tangent—and explaining their use. These functions are geometric quantities defined using the ratios of the Opposite, Adjacent, and Hypotenuse sides of the right triangle. Less common functions are the cosecant, secant, and cotangent functions. The history of the naming of the trigonometric functions is discussed along with an explanation of even more obscure functions: the versed sine, versed cosine, exsecant, and excosecant. The versed sine was used frequently in practical applications like astronomy, navigation, and surveying. Finally, inverse trigonometric functions and graphs of trigonometric functions are considered.

6. and beyond, to complex things

‘ … and beyond, to complex things’ first considers the Taylor series for the exponential function. One of the most famous, yet enigmatic, numbers in mathematics, e is an irrational number equal to 2.718281828. … Exponential functions deal with the phenomena of growth and decay. As calculus was starting to become established, curious parallels between the apparently disparate worlds of trigonometry and exponential functions were starting to appear. Imaginary numbers, Euler’s formula, and Euler’s identity are discussed along with the Argand diagram, De Moivre’s formula, hyperbolic trigonometric functions, and the catenary curve. Imaginary numbers are now at the heart of science and technology, and are used in the study of electromagnetic waves, cellular and wireless technologies, and fluid dynamics.

1. Why?

‘Why?’ considers some of the mathematical problems faced by scientists in the past: Hipparchus of Rhodes trying to predict the times of eclipses; Maurice Bressieu, the 16th-century French mathematician and humanist, calculating the height of a tower; and Lord Kelvin trying to predict ocean tide behaviour. Each of these scientists was faced with the same difficulty: the mathematical tools they had at their disposal were not up to the task of quantifying the phenomenon they were studying. This is the problem at the heart of trigonometry: how can we bring together geometry and computation to solve real physical problems? Today, examples of bridges between geometry and measurement are everywhere.

7. Spheres and more

‘Spheres and more’ considers the ten formulas for right-angled spherical triangles (and how they can be generated), the spherical Pythagorean theorem, and Napier’s rules. Spherical trigonometry was intended originally for astronomers, but medieval Islamic scholars used it to predict the beginning of the sacred month of Ramadan and the times of the five daily prayers. The impact of spherical geometry on Euclid’s axioms resulting in two types of non-Euclidean geometries—elliptical geometry and hyperbolic geometry—is also considered.

4. Identities, and more identities

The world of trigonometry is full of identities: some of them extremely useful, others beautiful, and a few that are simply bizarre. ‘Identities, and more identities’ takes a tour of the menagerie of identities, viewing a little from each of these categories. The first two examples are known as triangle identities, because they refer to angles and lengths in a given triangle. The Law of Sines and the Law of Cosines are discussed, along with Mollweide’s formulas, the Law of Tangents, Morrie’s Law, and the introduction of logarithms, which became the preferred computing tool in mathematical astronomy, and then in practical disciplines like surveying and architecture in the early 17th century.

3. Building a sine table with your bare hands

Technological advances, so pervasive in almost every aspect of our modern lives, become mundane to us almost overnight. How does a calculator find out, apparently effortlessly, that sin 33° = 0.5446? There are no right triangles drawn inside of the calculator, so where did that number come from? ‘Building a sine table with your bare hands’ looks at how different sine values have been calculated through history without any mechanical aids. It aintroduces the sine sum and difference laws, the half- and double-angle formulas, the golden triangle, and one of the most remarkable numbers in all of mathematics, the golden ratio, which appears in a dazzling array of areas of mathematics, natural phenomena, works of art, and architecture.