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Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an
Trigonometric_functions
Inverse functions of sin, cos, tan, etc.
and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used
Inverse trigonometric functions
Inverse_trigonometric_functions
Area of geometry, about angles and lengths
tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas
Trigonometry
study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (6th century AD), who discovered the sine function, cosine
History_of_trigonometry
Mathematical process of finding the derivative of a trigonometric function
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change
Differentiation of trigonometric functions
Differentiation_of_trigonometric_functions
functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions
List of integrals of trigonometric functions
List_of_integrals_of_trigonometric_functions
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for
List of trigonometric identities
List_of_trigonometric_identities
Fundamental trigonometric functions
In mathematics, sine and cosine are trigonometric functions. The sine and cosine of an acute angle are defined in the context of a right triangle: the
Sine_and_cosine
Hyperbolic analogues of trigonometric functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just
Hyperbolic_functions
Lists of values of mathematical functions
application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called
Trigonometric_table
Trigonometric values in terms of square roots and fractions
In mathematics, the values of the trigonometric functions can be expressed approximately, as in cos ( π / 4 ) ≈ 0.707 {\displaystyle \cos(\pi /4)\approx
Exact_trigonometric_values
Special function defined by an integral
In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions. The different sine integral definitions
Trigonometric_integral
Mathematical method in calculus
the function chosen to be dv. An alternative to this rule is the ILATE rule, where inverse trigonometric functions come before logarithmic functions. To
Integration_by_parts
Decomposition of periodic functions
periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum
Fourier_series
the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals. The inverse trigonometric functions are also known
List of integrals of inverse trigonometric functions
List_of_integrals_of_inverse_trigonometric_functions
Technique of integral evaluation
In mathematics, a trigonometric substitution replaces a trigonometric function with another expression. In calculus, trigonometric substitutions are a
Trigonometric_substitution
Rules for computing derivatives of functions
rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are functions of real numbers ( R {\textstyle \mathbb
Differentiation_rules
Important functions in solving differential equations
The trigonometric functions (especially sine and cosine) for complex square matrices occur in solutions of second-order systems of differential equations
Trigonometric functions of matrices
Trigonometric_functions_of_matrices
Mathematical notation based on the Arabic script
word of دالة زائدية "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to the way h
Modern Arabic mathematical notation
Modern_Arabic_mathematical_notation
Mathematical concept
notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions". A History of Mathematical
Inverse_function
functions List of integrals of irrational algebraic functions List of integrals of trigonometric functions List of integrals of inverse trigonometric
Lists_of_integrals
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Circle with radius of one
functions produce meaningful values for any real-valued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions –
Unit_circle
Use of complex numbers to evaluate integrals
involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e
Integration using Euler's formula
Integration_using_Euler's_formula
Relation between sine and cosine
Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions
Pythagorean trigonometric identity
Pythagorean_trigonometric_identity
Overview of and topical guide to trigonometry
theorem Trigonometric function Trigonometry of a tetrahedron Triangle (also see List of triangle topics) Sine, Cosine, Tangent (trigonometric function), Cotangent
Outline_of_trigonometry
Analytic function that does not satisfy a polynomial equation
logarithm and inverse trigonometric functions. All special functions such as the gamma, error, bessel, and Riemann zeta functions are transcendental. Equations
Transcendental_function
1 minus the cosine of an angle
versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Aryabhatiya, Section I) trigonometric tables. The versine of an
Versine
Type of mathematical function
polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions
Elementary_function
Mathematical approximation of a function
complex functions, such as logarithms, fractional powers, and inverse trigonometric functions, a principal branch is understood. The exponential function ex
Taylor_series
Change of variable for integrals involving trigonometric functions
integrals, which converts a rational function of trigonometric functions of x {\textstyle x} into an ordinary rational function of t {\textstyle t} by setting
Tangent half-angle substitution
Tangent_half-angle_substitution
List of values of a mathematical function
application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called
Mathematical_table
Branch of mathematics
and p-norm. While trigonometry deals with the relationships between angles and lengths in the plane using trigonometric functions defined relative to
Squigonometry
Geometric line segment whose endpoints lie on a circular arc
Commons has media related to Chord (geometry). History of Trigonometry Outline Trigonometric functions Archived 2017-03-10 at the Wayback Machine, focusing
Chord_(geometry)
Study of triangles in other spaces than the Euclidean plane
Polar/Trigonometric forms of hypercomplex numbers Polygonometry – trigonometric identities for multiple distinct angles The Lemniscate elliptic functions,
Generalized_trigonometry
Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics
Uses_of_trigonometry
Constant equal to twice pi
that the periodic trigonometric functions are simplified when using τ, as it corresponds to the period of the sine and cosine functions: sin repeats with
Tau_(mathematics)
Branch of mathematics studying functions of a complex variable
analytic. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately
Complex_analysis
Mathematical function
electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer
Jacobi_elliptic_functions
Collection of proofs of equations involving trigonometric functions
There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen
Proofs of trigonometric identities
Proofs_of_trigonometric_identities
Trigonometric functions introduced by Indian mathematicians and astronomers
Jyā, koṭi-jyā and utkrama-jyā are three trigonometric functions introduced by Indian mathematicians and astronomers. The earliest known Indian treatise
Jyā,_koti-jyā_and_utkrama-jyā
to the trigonometric functions. Inverse hyperbolic functions: inverses of the hyperbolic functions, analogous to the inverse circular functions. Logarithms:
List of mathematical functions
List_of_mathematical_functions
Association of one output to each input
defining the logarithm, the exponential and the trigonometric functions of a complex number. Functions whose domain are the nonnegative integers, known
Function_(mathematics)
Triangles without a right angle
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a
Acute_and_obtuse_triangles
Geometry of figures on the surface of a sphere
traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations
Spherical_trigonometry
Mathematical formula involving a given set of operations
Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the set
Closed-form_expression
Point to which functions converge in analysis
occur with rational functions. By noting that |x − p| represents a distance, the definition of a limit can be extended to functions of more than one variable
Limit_of_a_function
Relates the tangent of half of an angle to trigonometric functions of the entire angle
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half
Tangent_half-angle_formula
Development of the mathematical function
of tables of trigonometric functions and their natural logarithms. These tables greatly simplified calculations in spherical trigonometry, which are central
History_of_logarithms
Concept in mathematics
Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be
Walsh_function
Complex exponential in terms of sine and cosine
establishes the fundamental relationship between the trigonometric functions and the complex exponential function. It is named after Leonhard Euler, who mentioned
Euler's_formula
Non-sinusoidal waveform
{(4a)^{2}+p^{2}}}.} List of periodic functions Sine wave Square wave Sawtooth wave Pulse wave Sound Triangle function Wave Zigzag Kraft, Sebastian; Zölzer
Triangle_wave
Mathematical function, denoted exp(x) or e^x
distinguishing it from some other functions that are also commonly called exponential functions. These functions include the functions of the form f ( x ) = b
Exponential_function
Trigonometric function paired with another
sum to one right angle). This definition typically applies to trigonometric functions. The prefix "co-" can be found already in Edmund Gunter's Canon
Cofunction
Trigonometric function defined as secant minus one
external secant function (abbreviated exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function: exsec θ = sec
Exsecant
Equation that is satisfied for all values of the variables
of hyperbolic sines. The Gudermannian function gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involve
Identity_(mathematics)
Mathematical memory aids
In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions
Mnemonics_in_trigonometry
Shape with three sides
creates a new concept of trigonometric functions. The primary trigonometric functions are sine and cosine, as well as the other functions. They can be defined
Triangle
Course designed to prepare students for calculus
polynomials and rational functions are developed. Algebraic skills are exercised with trigonometric functions and trigonometric identities. The binomial
Precalculus
Mechanical analog computer
perform other calculations, such as square roots, exponentials, and trigonometric functions. The user may estimate the location of the decimal point in the
Slide_rule
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Simplification of the basic trigonometric functions
For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:
Small-angle_approximation
Coordinate system
in quotation marks are a mnemonic for remembering which three trigonometric functions (sine, cosine, tangent and their reciprocals) are positive in each
Quadrant_(plane_geometry)
Statistical transform
advantage is that calculating the trigonometric functions directly can be avoided. This is helpful when trigonometric functions are more expensive to compute
Box–Muller_transform
Iranian artist
Mathematical concepts he uses in his work include trigonometric functions, exponential function, Fibonacci sequence and the sawtooth wave. His artwork
Hamid_Naderi_Yeganeh
2005 book reformulating plane geometry
classical trigonometry. He also points out that, to a student with a scientific calculator, formulas that avoid square roots and trigonometric functions are
Divine Proportions: Rational Trigonometry to Universal Geometry
Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry
functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions
List_of_calculus_topics
Operation on mathematical functions
notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions". A History of Mathematical
Function_composition
Generalized mathematical function
roots, logarithms, and inverse trigonometric functions. To define a single-valued function from a complex multivalued function, one may distinguish one of
Multivalued_function
Book by Leonhard Euler
introduced the elementary transcendental functions, the logarithm, the exponential function, the trigonometric functions and their inverses without recourse
Introductio in analysin infinitorum
Introductio_in_analysin_infinitorum
Type of function
of the two sine functions vanishes. Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate
Orthogonal_functions
Indian mathematician and astronomer (1340–1425)
series, trigonometry, Spherical geometry, algebra, and analysis. He was the first to use infinite series approximations for a range of trigonometric functions
Madhava_of_Sangamagrama
Polynomial equation of degree 3
of trigonometric functions of angles related to 2 π / 7 {\displaystyle 2\pi /7} satisfy cubic equations. Given the cosine (or other trigonometric function)
Cubic_equation
Overview of and topical guide to geometry
Degree Minute Radian Circumference Diameter Trigonometric function Asymptotes Circular functions Periodic functions Law of cosines Law of sines Polar sine
Outline_of_geometry
Mathematical identity
hyperbolic functions are related to the inverse trigonometric functions similar to how the hyperbolic functions are related to the trigonometric functions, sin
Euler's continued fraction formula
Euler's_continued_fraction_formula
Arctangent function with two arguments
{\displaystyle \cos } , and tan {\displaystyle \tan } are trigonometric functions and x 2 + y 2 {\displaystyle \textstyle {\sqrt {x^{2}+y^{2}}}}
Atan2
Indefinite integral
functions are polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations under composition
Antiderivative
Development of mathematics in South Asia
Aryabhata's contributions include: Trigonometry: (See also : Aryabhata's sine table) Introduced the trigonometric functions. Defined the sine (jya) as the
Indian_mathematics
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
computer, is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials
CORDIC
for some of the following functions, though each function may have many equivalent definitions. All trigonometric functions listed have period 2 π {\displaystyle
List_of_periodic_functions
Mathematical statement which always holds true
expressions involving trigonometric functions need to be simplified. Another important application is the integration of non-trigonometric functions: a common technique
Law_(mathematics)
Hindu astronomy, mathematics, science school in India
mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called
Kerala school of astronomy and mathematics
Kerala_school_of_astronomy_and_mathematics
Method of evaluating certain integrals along paths in the complex plane
can be made to integrals involving trigonometric functions, so the integral is transformed into a rational function of a complex variable and then the
Contour_integration
Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions". Mathematics. 8 (687): 687. doi:10.3390/math8050687. Reynolds, Robert;
List_of_definite_integrals
Mathematical function, inverse of an exponential function
{1}{d}}\log _{10}c}.} Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions. Another critical
Logarithm
Triangle containing a 90-degree angle
hyperbolic functions as ratio of the sides of a right triangle, see the hyperbolic triangle of a hyperbolic sector. The values of the trigonometric functions can
Right_triangle
Calculator designed to calculate problems in science, engineering, and mathematics
multiplication, division) and advanced (trigonometric, hyperbolic, etc.) mathematical operations and functions. They have completely replaced slide rules
Scientific_calculator
trigonometry Sine Solution of triangles Spherical trigonometry Trigonometric functions Trigonometric substitution Trigonometric tables Trigonometry Uses
List_of_triangle_topics
Mathematical function relating circular and hyperbolic functions
{gd} \psi } . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s
Gudermannian_function
Mathematical functions
mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use:
Inverse_hyperbolic_functions
Special mathematical functions defined on the surface of a sphere
could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the
Spherical_harmonics
Swiss mathematician (1707–1783)
He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions. For any real number φ (taken to
Leonhard_Euler
Function equal to cos x + i sin x
"cis" is an acronym for "cos i sin". It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the
Cis_(mathematics)
Topics referred to by the same term
algebraic geometry secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciprocal) trigonometric function of the cosine the secant method
Secant
Number with a real and an imaginary part
the right. The series defining the real trigonometric functions sin and cos, as well as the hyperbolic functions sinh and cosh, also carry over to complex
Complex_number
Topics referred to by the same term
old trigonometric function cos−1y = cos−1(y), sometimes interpreted as arccos(y) or arccosine of y, the compositional inverse of the trigonometric function
Cos-1
Mathematical functions
\varpi } , as identities involving the trigonometric functions have analogues involving the lemniscate functions. For example, Viète's formula for π
Lemniscate_elliptic_functions
clerks to tabulate higher mathematical functions, such as exponential functions, logarithms, and trigonometric functions. These tables were eventually published
Mathematical_Tables_Project
)^{4}}}x^{2n}\end{aligned}}} The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series: ϑ 00 ( x
List_of_mathematical_series
Criterion for integration in terms of elementary functions
{\displaystyle \log |\sec x+\tan x|} , which uses only logarithms and trigonometric functions. More precisely, Liouville's theorem states that elementary antiderivatives
Liouville's theorem (differential algebra)
Liouville's_theorem_(differential_algebra)
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