In mathematics, the trigonometric functions or the circular functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray originating at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the length of the x-component (the adjacent of the angle or the run), and the tangent function gives the slope (y-component divided by the x-component). More precise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations.
http://en.wikipedia.org/wiki/Trigonometric_functions
Although trigonometry now is usually taught beginning with plane triangles, its origins lie in the world of astronomy and spherical triangles. Before the sixteenth century, astronomy was based on the notion that the earth stood at the center of a series of nested spheres. To calculate the positions of stars or planets, one needed to use concepts we now refer to as trigonometry
The earliest uses of trigonometric functions were related to the chords of a circle, and the recognition that the length of the chord subtended by a given angle x was (in modern terms) 2sin(x/2).
The Greek astronomer and mathematician Hipparchus produced the first known table of chords in 140 BC. His work was further developed by astronomers Menelaus (ca. AD 100) and Ptolemy (ca. AD 100), who relied on Babylonian observations and traditions
http://link.springer.com/chapter/10.1007%2F1-4020-2204-2_16
http://ualr.edu/lasmoller/trig.html
Dear Abedallah,
I think the most important reason is an antropologic one, namely the marvelling of the human mind in the presence of invariants in the universe. When humans began realizing that certain ratios were preserved invariant across different scales (think of pi, independent of the radius of a circle, the golden ratio, etc.), they began searching for more of these properties. And, as you mention, sines and cosines are very related to circles, so to me it comes as a natural consequence of the study of the circle and pi.
Cheers,
Fernando
I think that it is their periodicity that leads to their most important applications in engineering and sciences applications. Every phenomena that repeat as a function of time. Simple harmonic motion (sinusoidal motion), light and sound waves, electricity, gravitational waves in the universe, oscillations of the pendulum of a clock are periodic phenomena that are modeled mathematically by the trigonometric functions.
Trigonometric functions are used Fourier series which are widely applied in modeling of many engineering problems such as solving wave and heat partial equations. Fourier series approximate periodic Functions by Trigonometric Functions We used Fourier series in a publication for stochastic modeling of river water temperatures.
Article Predicting River Water temperatures using stochastic models:...
@Behrouz
Periodicity is a latter concept and most easily dealt with calculus. However, to my mind response of Fernando is a good explanation. With the discovery of pi early mathematician started search for search fixed values. In astronomy, in surveying early mathematician used these trigonometric functions. It is reported that Al-Beruni when returning from India, he on the top of a hill thought how to know distances and directions of cities in plain region. Later to know Qibla (direction of the Grand Masjid of Mecca) developed spherical trigonometry,
Dears @Fernando, @Behrouz, and @Mohammad, thank you for your contribution.
Dear Professor Abedallah M Rababah,
In engineering we realize that it is their periodicity that leads to their most important applications.
Although trigonometry now is usually taught beginning with plane triangles, its origins lie in the world of astronomy and spherical triangles. Before the sixteenth century, astronomy was based on the notion that the earth stood at the center of a series of nested spheres. To calculate the positions of stars or planets, one needed to use concepts we now refer to as trigonometry
The earliest uses of trigonometric functions were related to the chords of a circle, and the recognition that the length of the chord subtended by a given angle x was (in modern terms) 2sin(x/2).
The Greek astronomer and mathematician Hipparchus produced the first known table of chords in 140 BC. His work was further developed by astronomers Menelaus (ca. AD 100) and Ptolemy (ca. AD 100), who relied on Babylonian observations and traditions
http://link.springer.com/chapter/10.1007%2F1-4020-2204-2_16
http://ualr.edu/lasmoller/trig.html
Thank you dear Professors @Afaq and @Behrouz for the answers. I agree with you that the periodicity and the need for use in astronomy were very important reasons to develop the trigonometric functions.
This is a very good question.
In addition to what has already been incisively observed by others in this thread, there is the question of coherence in the study of trigonometry. The coherence question is considered in
K.C. Moore, Coherence, quantitative reasoning, and the trigonometry of students, University of Geogia:
http://www.uwyo.edu/wisdome/_files/documents/moore.pdf
It is pointed out that trigonometry lacks coherence in mathematics education (p. 75). That is, it should be clear how everyone should regard trigonometry. A coherent view of trigonometry starts with angle measure (p. 77).
More to the point, see
J.V. Sickle, A history of trigonometry education in the United States: 1776-1900, Ph.D. thesis, Columbia University, 2011:
file:///Users/jfpeters/Downloads/VanSickle_columbia_0054D_10129.pdf
It is pointed out that trigonometric functions came to be defined as ratios instead of line segments, due to advances in analytic geometry by Leonhard Euler and others in the 17th and 18th century (Abstract). For an overview of the history of trigonometry, see Chapter II, starting on page 32. One of the amazing things about this thesis is that it presents a topic that would normally be considered thoroughly understood and yet it is clear that there are many interesting questions about the definition of trigonometric functions that deserve a second look.
Dear @James, thank you for your contribution; the coherent view of the trigonometric functions is fundamental. The links are very helpful to understand the beginning and the origin of the definition of the trigonometric functions as ratios.
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