Uses of trigonometry

In Chapter XI of The Age of Reason, the American revolutionary and Enlightenment thinker Thomas Paine wrote:[1] From 1802 until 1871, the Great Trigonometrical Survey was a project to survey the Indian subcontinent with high precision.

Starting from the coastal baseline, mathematicians and geographers triangulated vast distances across the country.

One of the key achievements was measuring the height of Himalayan mountains, and determining that Mount Everest is the highest point on Earth.

[2] For the 25 years preceding the invention of the logarithm in 1614, prosthaphaeresis was the only known generally applicable way of approximating products quickly.

[4] The resemblance between the shape of a vibrating string and the graph of the sine function is no mere coincidence.

Many fields make use of trigonometry in more advanced ways than can be discussed in a single article.

Fourier series have a surprisingly diverse array of applications in many scientific fields, in particular in all of the phenomena involving seasonal periodicities mentioned above, and in wave motion, and hence in the study of radiation, of acoustics, of seismology, of modulation of radio waves in electronics, and of electric power engineering.

Among others are: the geometry of numbers, isoperimetric problems, recurrence of random walks, quadratic reciprocity, the central limit theorem, Heisenberg's inequality.

Fourier transforms involve integrals rather than sums, and are used in a similarly diverse array of scientific fields.

Trigonometric functions are also applied when statisticians study seasonal periodicities, which are often represented by Fourier series.