Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions)^[1]^[2] 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.

"Logarithmic sine" redirects here. For the Clausen-related function, see log sine function.

The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.

Notation[edit]

Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example $sin(x)$ . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression $\sin x+y$ would typically be interpreted to mean $\sin(x)+y,$ so parentheses are required to express $\sin(x+y).$

A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example $\sin ^{2}x$ and $\sin ^{2}(x)$ denote $\sin(x)\cdot \sin(x),$ not $\sin(\sin x).$ This differs from the (historically later) general functional notation in which $f^{2}(x)=(f\circ f)(x)=f(f(x)).$

However, the exponent ${-1}$ is commonly used to denote the inverse function, not the reciprocal. For example $\sin ^{-1}x$ and $\sin ^{-1}(x)$ denote the inverse trigonometric function alternatively written $\arcsin x\colon$ The equation $\theta =\sin ^{-1}x$ implies $\sin \theta =x,$ not $\theta \cdot \sin x=1.$ In this case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than ${-1}$ are not in common use.

Radians versus degrees[edit]

In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics).

However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions $sin$ and $cos$ can be defined for all complex numbers in terms of the exponential function, via power series,^[6] or as solutions to differential equations given particular initial values^[7] (see below), without reference to any geometric notions. The other four trigonometric functions ( $tan$ , $cot$ , $sec$ , $csc$ ) can be defined as quotients and reciprocals of $sin$ and $cos$ , except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle given in radians.^[6] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions.^[8] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.

When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 $π$ (≈ 6.28) rad. For real number x, the notations $sin x$ , $cos x$ , etc. refer to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., $sin x°$ , $cos x°$ , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/ $π$ )°, so that, for example, $sin π = sin 180°$ when we take x = $π$ . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = $π$ /180 ≈ 0.0175.

For an angle which, measured in degrees, is a multiple of three, the of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by ruler and compass.

exact trigonometric values

For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the of a non-real complex number. Galois theory allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.

cube root

For an angle which, expressed in degrees, is a , the sine and the cosine are algebraic numbers, which may be expressed in terms of $n$ th roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.

rational number

For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are . This is a corollary of Baker's theorem, proved in 1966.

transcendental numbers

$crd(θ) = 2 sin(θ / 2)$

$versin(θ) = 1 - cos(θ) = 2 sin 2 (θ / 2)$

$coversin(θ) = 1 - sin(θ) = versin(π / 2 - θ)$

$haversin(θ) = 1 / 2 versin(θ) = sin 2 (θ / 2)$

$exsec(θ) = sec(θ) - 1$

$excsc(θ) = exsec(π / 2 - θ) = csc(θ) - 1$

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) can be traced back to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.^[22] (See Aryabhata's sine table.)

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.^[23] With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant.^[23] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, and produced tables of tangents and cotangents.^[24]^[25] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.^[25] The trigonometric functions were later studied by mathematicians including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series.^[26] (See Madhava series and Madhava's sine table.)

The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.^[27]

The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).^[28]

The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie.^[29]

In a paper published in 1682, Gottfried Leibniz proved that $sin x$ is not an algebraic function of $x$ .^[30] Though introduced as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. He presented "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).^[22]

A few functions were common historically, but are now seldom used, such as the chord, the versine (which appeared in the earliest tables^[22]), the coversine, the haversine,^[31] the exsecant and the excosecant. The list of trigonometric identities shows more relations between these functions.

Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.^[32]^[33]^[34]^[35]

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Trigonometric functions"

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