Uses of Trigonometry

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Introduction to Trigonometry

The History and some Historical Uses of Trigonometry

If you use the links to the interactive exercises provided in the following brief history , think about the question first, and hover the mouse pointer
over the

icon for a hint, and then hover the mouse over the

icon for the answer.

You might like to visit the Right Triangle and the Trigonometric Functions page before attempting the exercises.

The ancient Babylonians, Egyptians, and Greeks used trigonometry to solve problems in astronomy, including the sizes of the sun and moon, and models of planetary motion. These problems were posed in spherical rather than plane spaces, but the principles of modern trigonometry evolved from those calculations. Hypparchus, a Greek of the 2nd Century BC, is generally regarded as the founder of trigonometry because he published the earliest known tables of trigonometric ratios.

These early mathematicians were more interested in chords of a circle than sides of triangles. The sine ratio first appeared in the work of Aryabhata, an Indian Hindu, in the 5th century AD, but he used the Sanskrit word "jiva" meaning a bowstring (consider the shape a chord and arc make on the circle). Jiva became jiba in translation by the Arabs, and jiba became jaib in later writings, which means fold in Arabic. This was translated into sinus, or fold in Latin, from where our modern sine is derived.

At first, the tangent was not associated with angles but was used for calculating heights from the length of cast shadows - also important for making sundials.
Thales, an ancient Greek philosopher, used the lengths of shadows to calculate the heights of pyramids.
The word tangent comes from the Latin word tangens meaning "to touch" (think of the tangent to a circle).

In 1533, the important German astronomer Regiomontanus published one of the first textbooks on trigonometry, "De triangulis omnimodis" (On Triangles). In the 16th Century, Rheticus, published "Trigonometry in Astronomy" which detailed the work of the astronomer Copernicus, and described the trigonometric functions in terms of triangles rather than circles. The secant function wasn't any use to the early astronomers and surveyors, but became important to the early navigators of the 15th century.
Secant comes from the Latin word secans meaning "cutting" (think of the secant line cutting the circle).

Modern trigonometry has uses beyond solving problems of circles and triangles. It can be used to describe any natural phenomenon that is periodic, and is useful in understanding abstract spaces in higher mathematics.

For more information on the history of trigonometry, try these links.	Brief History of Trigonometry History of Trigonometry Outline

Exercises for Historic Uses of Trigonometry

Chords of a Circle How might you use trigonometry to calculate the length of the chord AB of a circle with radius = r that is subtended by any known angle alpha? Return to Top

Lengths of Shadows How might Thules have calculated the height of an Egyptian pyramid using the length of its shadow? Return to Top

Navigation Early navigators used instruments, like the astrolobe and the sextant, which measured the angle from the horizon to the sun and other heavenly bodies, to calculate their latitude using trigonometric ratios. They also used trigonometry to position themselves relative to sightings made of landmarks. A complex type of trigonometry that deals with spherical angles was used in mapping the globe. For a more detiailed explanation of the role of trigonometry in navigation try these websites: The History of Navigation Celestial Navigation Map Projections Return to Top