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In Greek, trigonometry means “measuring three angles,” and it refers to the branch of mathematics that deals with angles, triangles, and the relationships between and among a triangle’s angles and sides. It includes the study of sines, cosines, and tangents, all of which are ratios of the sides of a triangle relative to a given angle.
The sine of an angle in a triangle is defined as the ratio of the side of the triangle opposite the angle to the hypotenuse of the triangle. Or, more simply, it is the measure of the opposite side divided by the measure of the hypotenuse.
The cosine of an angle in a triangle is defined as the ratio of the side of the triangle adjacent to the angle to the hypotenuse of the triangle. Or, more simply, it is the measure of the adjacent side divided by the measure of the hypotenuse.
The tangent of an angle in a triangle is defined as the ratio of the sine to the cosine. This can also be expressed as the ratio of the side of the triangle opposite the angle to the side of the triangle adjacent to the angle. Or, more simply, it is the measure of the opposite side divided by the measure of the adjacent side.
The art of trigonometry can be traced back at least as far as ancient Egypt, Mesopotamia, and the Indus Valley, which all used the basic functions of trigonometry at least as early as 2000 BC. In 499 CE the Indian mathematician Aryabhata laid down the first recorded tables for sine and cosine, which he called zya and kotizya respectively. These may well have been the origins of the words sine and cosine.
There are a few trigonometric identities that students should memorize for use in solving trigonometry problems:
sin2A + cos2A = 1
sin(A+B) = sinA cosB + cosA sinB
sin(A–B) = sinA cosB – cosA sinB
cos(A+B) = cosA cosB – sinA sinB
cos(A–B) = cosA cosB + sinA sinB
sin2A = 2sinAcosA
There are many more trigonometric identities that students learn as they advance in their trigonometry studies. [1]
Today trigonometry has a number of important uses. Trigonometry is used to measure the distance to nearby stars through the process of triangulation. Geographers and navigators use it to find distances between landmarks.
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[1] Louis Valdez-Sanchez, “Table of Trigonometric Identities,” 3 December 1996, <http://www.sosmath.com/trig/Trig5/trig5/trig5.html> (25 September 2006).
