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Trigonometry |
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Reference |
Laws and theorems |
Calculus |
Mathematicians |
Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle'and μέτρον (métron) 'measure') [1] is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. [2] The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. [3]
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. [4]
Trigonometry is known for its many identities. These trigonometric identities [5] are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. [6]
Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. [8] They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method. [9]
In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry. [10] In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest . [11] Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. [12] (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.
The modern definition of the sine is first attested in the Surya Siddhanta , and its properties were further documented in the 5th century (AD) by Indian mathematician and astronomer Aryabhata. [13] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. In 830 AD, Persian mathematician Habash al-Hasib al-Marwazi produced the first table of cotangents. [14] [15] By the 10th century AD, in the work of Persian mathematician Abū al-Wafā' al-Būzjānī, all six trigonometric functions were used. [16] Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. [16] He also made important innovations in spherical trigonometry [17] [18] [19] The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. [20] [21] [22] He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. [15] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. [23] Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi. [24] One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, who was encouraged to write, and provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years. [25] At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond. [26] Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. [27] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. [28] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. [29] Also in the 18th century, Brook Taylor defined the general Taylor series. [30]
Trigonometric ratios are the ratios between edges of a right triangle. These ratios depend only on one acute angle of the right triangle, since any two right triangles with the same acute angle are similar. [31]
So, these ratios define functions of this angle that are called trigonometric functions. Explicitly, they are defined below as functions of the known angle A, where a, b and h refer to the lengths of the sides in the accompanying figure.
In the following definitions, 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. See below under Mnemonics.
The reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". [32]
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. [33] These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.
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 and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA: [34]
One way to remember the letters is to sound them out phonetically (i.e. /ˌsoʊkəˈtoʊə/ SOH-kə-TOH-ə, similar to Krakatoa). [35] Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid". [36]
Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. [37] In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where and . [37] This representation allows for the calculation of commonly found trigonometric values, such as those in the following table: [38]
Function | 0 | ||||||||
---|---|---|---|---|---|---|---|---|---|
sine | |||||||||
cosine | |||||||||
tangent | undefined | ||||||||
secant | undefined | ||||||||
cosecant | undefined | undefined | |||||||
cotangent | undefined | undefined |
Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments [39] (see trigonometric function).
The following table summarizes the properties of the graphs of the six main trigonometric functions: [40] [41]
Function | Period | Domain | Range | Graph |
---|---|---|---|---|
sine | ![]() | |||
cosine | ![]() | |||
tangent | ![]() | |||
secant | ![]() | |||
cosecant | ![]() | |||
cotangent | ![]() |
Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible. [42] : 48ff
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table: [42] : 48ff [43] : 521ff
Name | Usual notation | Definition | Domain of x for real result | Range of usual principal value (radians) | Range of usual principal value (degrees) |
---|---|---|---|---|---|
arcsine | y = arcsin(x) | x = sin(y) | −1 ≤ x ≤ 1 | −π/2 ≤ y ≤ π/2 | −90° ≤ y ≤ 90° |
arccosine | y = arccos(x) | x = cos(y) | −1 ≤ x ≤ 1 | 0 ≤ y ≤ π | 0° ≤ y ≤ 180° |
arctangent | y = arctan(x) | x = tan(y) | all real numbers | −π/2 < y < π/2 | −90° < y < 90° |
arccotangent | y = arccot(x) | x = cot(y) | all real numbers | 0 < y < π | 0° < y < 180° |
arcsecant | y = arcsec(x) | x = sec(y) | x ≤ −1 or 1 ≤ x | 0 ≤ y < π/2 or π/2 < y ≤ π | 0° ≤ y < 90° or 90° < y ≤ 180° |
arccosecant | y = arccsc(x) | x = csc(y) | x ≤ −1 or 1 ≤ x | −π/2 ≤ y < 0 or 0 < y ≤ π/2 | −90° ≤ y < 0° or 0° < y ≤ 90° |
When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations: [44]
With these definitions the trigonometric functions can be defined for complex numbers. [45] When extended as functions of real or complex variables, the following formula holds for the complex exponential:
This complex exponential function, written in terms of trigonometric functions, is particularly useful. [46] [47]
Trigonometric functions were among the earliest uses for mathematical tables. [48] 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. [49] Slide rules had special scales for trigonometric functions. [50]
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). [51] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. [52] The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions. [53]
In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables [54] ), the coversine (coversin(θ) = 1 − sin(θ) = versin(π/2 − θ)), the haversine (haversin(θ) = 1/2versin(θ) = sin2(θ/2)), [55] the exsecant (exsec(θ) = sec(θ) − 1), and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). See List of trigonometric identities for more relations between these functions.
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, [56] predicting eclipses, and describing the orbits of the planets. [57]
In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars, [58] as well as in satellite navigation systems. [19]
Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation. [59]
Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles. [60]
In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects. [61]
On a larger scale, trigonometry is used in geography to measure distances between landmarks. [62]
The sine and cosine functions are fundamental to the theory of periodic functions, [63] such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.
Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics [64] and communications, [65] among other fields.
Trigonometry is useful in many physical sciences, [66] including acoustics, [67] and optics. [67] In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems. [68]
Other fields that use trigonometry or trigonometric functions include music theory, [69] geodesy, audio synthesis, [70] architecture, [71] electronics, [69] biology, [72] medical imaging (CT scans and ultrasound), [73] chemistry, [74] number theory (and hence cryptology), [75] seismology, [67] meteorology, [76] oceanography, [77] image compression, [78] phonetics, [79] economics, [80] electrical engineering, mechanical engineering, civil engineering, [69] computer graphics, [81] cartography, [69] crystallography [82] and game development. [81]
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. [83]
Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities, [84] relate both the sides and angles of a given triangle.
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 (as shown in the diagram).
The law of sines (also known as the "sine rule") for an arbitrary triangle states: [85]
where is the area of the triangle and R is the radius of the circumscribed circle of the triangle:
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: [85]
or equivalently:
The law of tangents , developed by François Viète, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables. [86] It is given by:
Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides: [85]
The following trigonometric identities are related to the Pythagorean theorem and hold for any value: [87]
The second and third equations are derived from dividing the first equation by and , respectively.
Euler's formula, which states that , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities. [31]
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles, while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals; The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data and the technique gives two possible values for the enclosed angle.
In geometry, a hypotenuse is the side of a right triangle opposite the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called catheti or legs. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. Mathematically, this can be written as , where a is the length of one leg, b is the length of another leg, and c is the length of the hypotenuse.
In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B produce the same value for all values of the variables within a certain domain of discourse. In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined. For example, and are identities. Identities are sometimes indicated by the triple bar symbol ≡ instead of =, the equals sign. Formally, an identity is a universally quantified equality.
In trigonometry, the law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles.
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices.
For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:
In spherical trigonometry, the law of cosines is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted as and .
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function, cosine function, and versine function.
There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles and the ratio between their sides. The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.
In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.
In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions.
One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
His major contribution in mathematics (Nasr, 1996, pp. 208–214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.