trigonometry

Dictionary: trig·o·nom·e·try (trĭg'ə-nŏm'ĭ-trē) pronunciation

n.
The branch of mathematics that deals with the relationships between the sides and the angles of triangles and the calculations based on them, particularly the trigonometric functions.

[New Latin trigōnometria : Greek trigōnon, triangle; see trigon + Greek -metriā, -metry.]

trigonometric trig'o·no·met'ric (-nə-mĕt'rĭk) or trig'o·no·met'ri·cal (-rĭ-kəl) adj.
trigonometrically trig'o·no·met'ri·cal·ly adv.

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Britannica Concise Encyclopedia: trigonometry

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Mathematical discipline dealing with the relationships between the sides and angles of triangles. Literally, it means triangle measurement, though its applications extend far beyond geometry. It emerged as a rigorous discipline in the 15th century, when the demand for accurate surveying techniques and navigational methods led to its use for the "solution" of right triangles, or the calculation of the lengths of two sides of a right triangle given one of its acute angles and the length of one side. The solution can be found by using ratios in the form of the trigonometric functions.

For more information on trigonometry, visit Britannica.com.

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The branch of mathematics dealing with the relationships between the angles and the sides of triangles, and the calculations based on them. It originated as the study of certain mathematical relations in a triangle containing a right angle (90°).

Columbia Encyclopedia: trigonometry

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trigonometry [Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy.

The Basic Trigonometric Functions

Trigomometry originated as the study of certain mathematical relations originally defined in terms of the angles and sides of a right triangle, i.e., one containing a right angle (90°). Six basic relations, or trigonometric functions, are defined.

The trigonometric functions are defined in terms of the angles and sides of a right triangle.

If A, B, and C are the measures of the angles of a right triangle (C=90°) and a, b, and c are the lengths of the respective sides opposite these angles, then the six functions are expressed for one of the acute angles, say A, as various ratios of the opposite side (a), the adjacent side (b), and the hypotenuse (c), as set out in the table.

Although the actual lengths of the sides of a right triangle may have any values, the ratios of the lengths will be the same for all similar right triangles, large or small; these ratios depend only on the angles and not on the actual lengths. The functions occur in pairs-sine and cosine, tangent and cotangent, secant and cosecant-called cofunctions. In equations they are usually represented as sin, cos, tan, cot, sec, and csc. Since in ordinary (Euclidean) plane geometry the sum of the angles of a triangle is 180°, angles A and B must add up to 90° and therefore are complementary angles. From the definitions of the functions, it may be seen that sin B=cos A, cos B=sin A, tan B=cot A, and sec B=csc A; in general, the function of an angle is equal to the cofunction of its complement. Since the hypotenuse (c), is always the longest side of a right triangle, the values of the sine and cosine are always between zero and one, the values of the secant and cosecant are always equal to or greater than one, and the values of the tangent and cotangent are unbounded, increasing from zero without limit.

For certain special right triangles the values of the functions may be calculated easily; e.g., in a right triangle whose acute angles are 30° and 60° the sides are in the ratio 1 :√3 : 2, so that sin 30°=cos 60°=1/2, cos 30°=sin 60°=√3/2, tan 30°=cot 60°=1/√3, cot 30°=tan 60°=√3, sec 30°=csc 60°=2/√3, and csc 30°=sec 60°=2. For other angles, the values of the trigonometric functions are usually found from a set of tables or a scientific calculator. For the limiting values of 0° and 90°, the length of one side of the triangle approaches zero while the other approaches that of the hypotenuse, resulting in the values sin 0°=cos 90°=0, cos 0°=sin 90°=1, tan 0°=cot 90°=0, and sec 0°=csc 90°=1; since division by zero is undefined, cot 0°, tan 90°, csc 0°, and sec 90° are all undefined, having infinitely large values.

A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle. For example, in any plane triangle a/sin A=b/sin B=c/sin C. This relationship is known as the Law of Sines. The related Law of Cosines holds that a²=b²+c²−2bc cosA and the Law of Tangents holds that (a−b)/(a+b)=[tan ¹/₂(A−B)]/[tan ¹/₂(A+B)]. Each of the trigonometric functions can be represented by an infinite series.

Extension of the Trigonometric Functions

The notion of the trigonometric functions can be extended beyond 90° by defining the functions with respect to Cartesian coordinates. Let r be a line of unit length from the origin to the point P (x,y), and let θ be the angle r makes with the positive x-axis. The six functions become sin θ =y/r=y, cos θ=x/r=x, tan θ=y/x, cot θ=x/y, sec θ=r/x=1/x, and csc θ=r/y=1/y. As θ increases beyond 90°, the point P crosses the y-axis and x becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180°, P is in quadrant III, y is also negative, and only tan θ and cot θ are positive, while beyond θ=270° P moves into quadrant IV, x becomes positive again, and cos θ and sec θ are positive.

The trigonometric functions of the angle formed by the x-axis and the line r terminating at point P may be expressed in terms of r and the x- and y-coordinates of P. For θ₁ both x and y are positive; for θ₂ x is negative.

Since the positions of r for angles of 360° or more coincide with those already taken by r as θ increased from 0°, the values of the functions repeat those taken between 0° and 360° for angles greater than 360°, repeating again after 720°, and so on.

Graph of y = sin θ as a function of the angle θ. The values of sin θ repeat every 360°.

This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.

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IN BRIEF: The mathematics of tridimensional shapes.

Did you study geometry and trigonometry at the university?

Wikipedia: Trigonometry

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"Trig" redirects here. For other uses, see Trig (disambiguation).

The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of the trigonometric functions of those angles.

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

Trigonometry (from Greek trigōnon "triangle" + metron "measure")^[1] is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships, as well as describing angles in general and the motion of waves such as sound and light waves.

Trigonometry is usually taught in secondary schools either as a separate course or as part of a precalculus course. It has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation.

1 History
2 Overview
3 Applications of trigonometry
4 Common formulas
5 See also
6 Notes
7 References
8 External links

History

Main article: History of trigonometry

Pre-Hellenic societies such as the ancient Egyptians and Babylonians lacked the concept of an angle measure, but they studied the ratios of the sides of similar triangles and discovered some properties of these ratios. Ancient Greek mathematicians such as Euclid and Archimedes studied the properties of the chord of an angle and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. The sine function in its modern form was first defined in the Surya Siddhanta and its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata.^[2] These Indian works were translated and expanded by medieval Islamic scholars. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry. At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.^[3] One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little known in 16th century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explaining its basic concepts.

Overview

In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$

The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$

The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.$

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).

The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.

With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and an angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.

Graphing process of y = sin(x) using a unit circle.

Graphing process of y = tan(x) using a unit circle.

Graphing process of y = csc(x) using a unit circle.

Extending the definitions

Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians.

The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals.

The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex function cis is particularly useful

$\operatorname{cis}\,x = \cos x + i\sin x \! = e^{ix}.$

See Euler's and De Moivre's formulas.

Mnemonics

A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, as in SOH-CAH-TOA.

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

The memorization of this mnemonic can be aided by expanding it into a phrase, such as "Some Officers Have Curly Auburn Hair Till Old Age".^[4] Any memorable phrase constructed of words beginning with the letters S-O-H-C-A-H-T-O-A will serve.

Calculating trigonometric functions

Main article: Generating trigonometric tables

Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.

Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.

Applications of trigonometry

Main article: Uses of trigonometry

There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.

Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

Marine sextants like this are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can then be determined from several such measurements.

Common formulas

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. There are some identities which equate an expression to a different expression involving the same angles and these are listed in List of trigonometric identities, and then there are the triangle identities which relate the sides and angles of a given triangle and these are listed below.

Triangle with sides a,b,c respectively opposite angles A,B,C, as described to the left

In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles.

Law of sines

The law of sines (also known as the "sine rule") for an arbitrary triangle states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,$

where R is the radius of the circumcircle of the triangle:

$R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.$

Another law involving sines can be used to calculate the area of a triangle. If you know two sides and the angle between the sides, the area of the triangle becomes:

$\mbox{Area} = \frac{1}{2}a b\sin C.$

Law of cosines

The law of cosines ( known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:

$c^2=a^2+b^2-2ab\cos C ,\,$

or equivalently:

$\cos C=\frac{a^2+b^2-c^2}{2ab}.\,$

Law of tangents

The law of tangents:

$\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}$

Notes

^ "trigonometry". Online Etymology Dictionary. http://www.etymonline.com/index.php?search=trigonometry.
^ Boyer p215
^ Boyer p237, p274
^ "What does SOHCAHTOA stand for?". Acronym Finder. http://www.acronymfinder.com/Some-Officers-Have-Curly-Auburn-Hair-Till-Old-Age-(mnemonic-for-remembering-sine,-cosine-and-tangent)-(SOHCAHTOA).html. Retrieved 2008-10-27.

References

Boyer, Carl B. (1991). A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. ISBN 0471543977.
Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press.
Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram MathWorld.

External links

Find more about Trigonometry on Wikipedia's sister projects:

Definitions from Wiktionary

Textbooks from Wikibooks

Quotations from Wikiquote

Source texts from Wikisource

Images and media from Commons

News stories from Wikinews

Learning resources from Wikiversity

Trigonometric Delights, by Eli Maor, Princeton University Press, 1998. Ebook version, in PDF format, full text presented.
Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
Trigonometry on Mathwords.com index of trigonometry entries on Mathwords.com
Benjamin Banneker's Trigonometry Puzzle at Convergence
Dave's Short Course in Trigonometry by David Joyce of Clark University

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Translations: Trigonometry

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Dansk (Danish)
n. - trigonometri

Nederlands (Dutch)
trigonometrie

Français (French)
n. - trigonométrie

Deutsch (German)
n. - Trigonometrie

Ελληνική (Greek)
n. - (μαθημ.) τριγωνομετρία

Italiano (Italian)
trigonometria

Português (Portuguese)
n. - trigonometria (f)

Русский (Russian)
тригонометрия

Español (Spanish)
n. - trigonometría

Svenska (Swedish)
n. - trigonometri

中文（简体）(Chinese (Simplified))
三角, 三角法

中文（繁體）(Chinese (Traditional))
n. - 三角, 三角法

한국어 (Korean)
n. - 삼각법, 삼각술

日本語 (Japanese)
n. - 三角法

العربيه (Arabic)
‏(الاسم) ألمثلثات, علم ألمثلثات‏

עברית (Hebrew)
n. - ‮טריגונומטריה‬

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