Trigonometric functions

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles.

Modern definitions express trigonometric functions as infinite series or as solutions of differential equations.

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.

[5] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions.

[7] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.

radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A.

The other trigonometric functions can be found along the unit circle as By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is Since a rotation of an angle of

By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that

That is, the equalities hold for any angle θ and any integer k. The algebraic expressions for the most important angles are as follows: Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.

G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number.

[16] More precisely, defining one has the following series expansions:[17] The following continued fractions are valid in the whole complex plane: The last one was used in the historically first proof that π is irrational.

[20] Combining the (–n)th with the nth term lead to absolutely convergent series: Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: The following infinite product for the sine is due to Leonhard Euler, and is of great importance in complex analysis:[21] This may be obtained from the partial fraction decomposition of

Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups.

of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line.

: Together with this is the tangent half-angle substitution, which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.

The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule.

Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series.

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 defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).

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.

[33] 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.

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

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

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

The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748).

[30] A few functions were common historically, but are now seldom used, such as the chord, versine (which appeared in the earliest tables[30]), haversine, coversine,[39] half-tangent (tangent of half an angle), and exsecant.

[40][41][42][43] The word sine derives[44] from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin.

[45] The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".

Basis of trigonometry: if two right triangles have equal acute angles , they are similar , so their corresponding side lengths are proportional .
In this right triangle, denoting the measure of angle BAC as A: sin A = ⁠ a / c ⁠ ; cos A = ⁠ b / c ⁠ ; tan A = ⁠ a / b ⁠ .
Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians . The points labeled 1 , Sec( θ ) , Csc( θ ) represent the length of the line segment from the origin to that point. Sin( θ ) , Tan( θ ) , and 1 are the heights to the line starting from the x -axis, while Cos( θ ) , 1 , and Cot( θ ) are lengths along the x -axis starting from the origin.
Top: Trigonometric function sin θ for selected angles θ , π − θ , π + θ , and 2 π − θ in the four quadrants.
Bottom: Graph of sine versus angle. Angles from the top panel are identified.
All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O .
Sine function on unit circle (top) and its graph (bottom)
In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle . The y -axis ordinates of A , B and D are sin θ , tan θ and csc θ , respectively, while the x -axis abscissas of A , C and E are cos θ , cot θ and sec θ , respectively.
Signs of trigonometric functions in each quadrant. Mnemonics like " all s tudents t ake c alculus" indicates when s ine, c osine, and t angent are positive from quadrants I to IV. [ 8 ]
Trigonometric functions: Sine , Cosine , Tangent , Cosecant (dotted) , Secant (dotted) , Cotangent (dotted) – animation
The unit circle , with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.
Graphs of sine, cosine and tangent
The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.
Animation for the approximation of cosine via Taylor polynomials.
together with the first Taylor polynomials
and are the real and imaginary part of respectively.
A Lissajous curve , a figure formed with a trigonometry-based function.
An animation of the additive synthesis of a square wave with an increasing number of harmonics
Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental ( k = 1 ) have additional nodes. The oscillation seen about the sawtooth when k is large is called the Gibbs phenomenon .