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trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
, trigonometric identities are equalities that involve
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.


Pythagorean identities

The basic relationship between the sine and cosine is given by the Pythagorean identity: :\sin^2\theta + \cos^2\theta = 1, where \sin^2 \theta means (\sin \theta)^2 and \cos^2 \theta means (\cos \theta)^2. This can be viewed as a version of the
Pythagorean theorem 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 (the side opposit ...
, and follows from the equation x^2 + y^2 = 1 for the
unit circle 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 (0, 0) in the Cartesian coordinate system in the Eucli ...
. This equation can be solved for either the sine or the cosine: \begin \sin\theta &= \pm \sqrt, \\ \cos\theta &= \pm \sqrt. \end where the sign depends on the quadrant of \theta. Dividing this identity by \sin^2 \theta, \cos^2 \theta, or both yields the following identities: \begin &1 + \cot^2\theta = \csc^2\theta \\ &\tan^2\theta + 1 = \sec^2\theta \\ &\sec^2\theta + \csc^2\theta = \sec^2\theta\csc^2\theta \end Using these identities, it is possible to express any trigonometric function in terms of any other (
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
a plus or minus sign):


Reflections, shifts, and periodicity

By examining the unit circle, one can establish the following properties of the trigonometric functions.


Reflections

When the direction of a Euclidean vector is represented by an angle \theta, this is the angle determined by the free vector (starting at the origin) and the positive x-unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive x-axis. If a line (vector) with direction \theta is reflected about a line with direction \alpha, then the direction angle \theta^ of this reflected line (vector) has the value \theta^ = 2 \alpha - \theta. The values of the trigonometric functions of these angles \theta,\;\theta^ for specific angles \alpha satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as .


Shifts and periodicity


Signs

The sign of trigonometric functions depends on quadrant of the angle. If < \theta \leq \pi and is the
sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avo ...
, \begin \sgn(\sin \theta) &= \begin +1 & \text\ \ 0 < \theta < \pi \\ -1 & \text\ \ < \theta < 0 \\ 0 & \text\ \ \theta \in \ \end \\ mu\sgn(\cos \theta) &= \begin +1 & \text\ \ < \theta < \tfrac12\pi \\ -1 & \text\ \ < \theta < -\tfrac12\pi \ \ \text\ \ \tfrac12\pi < \theta < \pi\\ 0 & \text\ \ \theta \in \bigl\ \end \\ mu\sgn(\tan \theta) &= \begin +1 & \text\ \ < \theta < -\tfrac12\pi \ \ \text\ \ 0 < \theta < \tfrac12\pi \\ -1 & \text\ \ < \theta < 0 \ \ \text\ \ \tfrac12\pi < \theta < \pi \\ 0 & \text\ \ \theta \in \bigl\ \end \\ mu\sgn(\csc \theta) &= \begin +1 & \text\ \ 0 < \theta < \pi \\ -1 & \text\ \ < \theta < 0 \\ \text & \text\ \ \theta \in \ \end \\ mu\sgn(\sec \theta) &= \begin +1 & \text\ \ < \theta < \tfrac12\pi \\ -1 & \text\ \ < \theta < -\tfrac12\pi \ \ \text\ \ \tfrac12\pi < \theta < \pi\\ \text & \text\ \ \theta \in \bigl\ \end \\ mu\sgn(\cot \theta) &= \begin +1 & \text\ \ < \theta < -\tfrac12\pi \ \ \text\ \ 0 < \theta < \tfrac12\pi \\ -1 & \text\ \ < \theta < 0 \ \ \text\ \ \tfrac12\pi < \theta < \pi \\ \text & \text\ \ \theta \in \bigl\ \end \end The trigonometric functions are periodic with common period 2\pi, so for values of outside the interval (, \pi], they take repeating values (see above).


Angle sum and difference identities

These are also known as the (or ). \begin \sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \sin(\alpha - \beta) &= \sin \alpha \cos \beta - \cos \alpha \sin \beta \\ \cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \cos(\alpha - \beta) &= \cos \alpha \cos \beta + \sin \alpha \sin \beta \end The angle difference identities for \sin(\alpha - \beta) and \cos(\alpha - \beta) can be derived from the angle sum versions by substituting -\beta for \beta and using the facts that \sin(-\beta) = -\sin(\beta) and \cos(-\beta) = \cos(\beta). They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here. These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.


Sines and cosines of sums of infinitely many angles

When the series \sum_^\infty \theta_i absolute convergence, converges absolutely then :\sin\left(\sum_^\infty \theta_i\right) =\sum_ (-1)^\frac \sum_ \left(\prod_ \sin\theta_i \prod_ \cos\theta_i\right) :\cos\left(\sum_^\infty \theta_i\right) =\sum_ ~ (-1)^\frac ~~ \sum_ \left(\prod_ \sin\theta_i \prod_ \cos\theta_i\right) \,. Because the series \sum_^\infty \theta_i converges absolutely, it is necessarily the case that \lim_ \theta_i = 0, \lim_ \sin \theta_i = 0, and \lim_ \cos \theta_i = 1. In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles \theta_i are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.


Tangents and cotangents of sums

Let e_k (for k = 0, 1, 2, 3, \ldots) be the th-degree elementary symmetric polynomial in the variables x_i = \tan \theta_i for i = 0, 1, 2, 3, \ldots, that is, : \begin e_0 & = 1 \\ pte_1 & = \sum_i x_i & & = \sum_i \tan\theta_i \\ pte_2 & = \sum_ x_i x_j & & = \sum_ \tan\theta_i \tan\theta_j \\ pte_3 & = \sum_ x_i x_j x_k & & = \sum_ \tan\theta_i \tan\theta_j \tan\theta_k \\ & \ \ \vdots & & \ \ \vdots \end Then :\begin\tan\left(\sum_i \theta_i\right) & = \frac \\& = \frac = \frac \\ \cot\left(\sum_i \theta_i\right) & = \frac \end using the sine and cosine sum formulae above. The number of terms on the right side depends on the number of terms on the left side. For example: :\begin \tan(\theta_1 + \theta_2) & = \frac = \frac = \frac, \\ pt\tan(\theta_1 + \theta_2 + \theta_3) & = \frac = \frac, \\ pt\tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) & = \frac \\ pt& = \frac, \end and so on. The case of only finitely many terms can be proved by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
.


Secants and cosecants of sums

:\begin \sec\left(\sum_i \theta_i\right) & = \frac \\ pt\csc\left(\sum_i \theta_i \right) & = \frac \end where e_k is the th-degree elementary symmetric polynomial in the variables x_i = \tan \theta_i, i = 1, \ldots, n, and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left. The case of only finitely many terms can be proved by mathematical induction on the number of such terms. For example, :\begin \sec(\alpha+\beta+\gamma) & = \frac \\ pt\csc(\alpha+\beta+\gamma) & = \frac. \end


Ptolemy's theorem

Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved (see the section on classical antiquity in the page History of trigonometry). It states that in a cyclic quadrilateral ABCD, as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities. The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here. By Thales's theorem, \angle DAB and \angle DCB are both right angles. The right-angled triangles DAB and DCB both share the hypotenuse \overline of length 1. Thus, the side \overline = \sin \alpha, \overline = \cos \alpha, \overline = \sin \beta and \overline = \cos \beta. By the
inscribed angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an in ...
theorem, the central angle subtended by the chord \overline at the circle's center is twice the angle \angle ADC, i.e. 2(\alpha + \beta). Therefore, the symmetrical pair of red triangles each has the angle \alpha + \beta at the center. Each of these triangles has a hypotenuse of length \frac, so the length of \overline is 2 \times \frac \sin(\alpha + \beta), i.e. simply \sin(\alpha + \beta). The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also \sin(\alpha + \beta). When these values are substituted into the statement of Ptolemy's theorem that , \overline, \cdot , \overline, =, \overline, \cdot , \overline, +, \overline, \cdot , \overline, , this yields the angle sum trigonometric identity for sine: \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta . The angle difference formula for \sin(\alpha - \beta) can be similarly derived by letting the side \overline serve as a diameter instead of \overline.


Multiple-angle formulae


Multiple-angle formulae


Double-angle formulae

Formulae for twice an angle. :\sin (2\theta) = 2 \sin \theta \cos \theta = (\sin \theta +\cos \theta)^2 - 1 = \frac :\cos (2\theta) = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta = \frac :\tan (2\theta) = \frac :\cot (2\theta) = \frac = \frac :\sec (2\theta) = \frac = \frac :\csc (2\theta) = \frac = \frac


Triple-angle formulae

Formulae for triple angles. :\sin (3\theta) =3\sin\theta - 4\sin^3\theta = 4\sin\theta\sin\left(\frac -\theta\right)\sin\left(\frac + \theta\right) :\cos (3\theta) = 4 \cos^3\theta - 3 \cos\theta =4\cos\theta\cos\left(\frac -\theta\right)\cos\left(\frac + \theta\right) :\tan (3\theta) = \frac = \tan \theta\tan\left(\frac - \theta\right)\tan\left(\frac + \theta\right) :\cot (3\theta) = \frac :\sec (3\theta) = \frac :\csc (3\theta) = \frac


Multiple-angle and half-angle formulae

:\begin \sin(n\theta) &= \sum_ (-1)^\frac \cos^ \theta \sin^k \theta = \sin\theta\sum_^\sum_^ (-1)^ \cos^ \theta \\ &=2^ \prod_^ \sin(k\pi/n+\theta)\\ \cos(n\theta) &= \sum_ (-1)^\frac \cos^ \theta \sin^k \theta = \sum_^\sum_^ (-1)^ \cos^ \theta\\ \cos((2n+1)\theta)&=(-1)^n 2^\prod_^\cos(k\pi/(2n+1)-\theta)\\ \cos(2 n \theta)&=(-1)^n 2^ \prod_^ \cos((1+2k)\pi/(4n)-\theta) \end : \tan(n\theta) = \frac


Chebyshev method

The
Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...
method is a recursive
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for finding the th multiple angle formula knowing the (n-1)th and (n-2)th values. :\cos(nx) can be computed from \cos((n-1)x), \cos((n-2)x), and \cos(x) with :\cos(nx)=2 \cos x \cos((n-1)x) - \cos((n-2)x). This can be proved by adding together the formulae :\cos ((n-1)x + x) = \cos ((n-1)x) \cos x-\sin ((n-1)x) \sin x :\cos ((n-1)x - x) = \cos ((n-1)x) \cos x+\sin ((n-1)x) \sin x It follows by induction that \cos(nx) is a polynomial of \cos x, the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition. Similarly, \sin(nx) can be computed from \sin((n-1)x), \sin((n-2)x), and with :\sin(nx)=2 \cos x \sin((n-1)x)-\sin((n-2)x) This can be proved by adding formulae for \sin((n-1)x+x) and \sin((n-1)x-x). Serving a purpose similar to that of the Chebyshev method, for the tangent we can write: :\tan (nx) = \frac\,.


Half-angle formulae

\begin \sin \frac &= \sgn\left(\sin\frac\theta2\right) \sqrt \\ pt \cos \frac &= \sgn\left(\cos\frac\theta2\right) \sqrt \\ pt \tan \frac &= \frac = \frac = \csc \theta - \cot \theta = \frac \\ mu &= \sgn(\sin \theta) \sqrt\frac = \frac \\ pt \cot \frac &= \frac = \frac = \csc \theta + \cot \theta = \sgn(\sin \theta) \sqrt\frac \\ \sec \frac &= \sgn\left(\cos\frac\theta2\right) \sqrt \\ \csc \frac &= \sgn\left(\sin\frac\theta2\right) \sqrt \\ \end Also \begin \tan\frac &= \frac \\ pt \tan\left(\frac + \frac\right) &= \sec\theta + \tan\theta \\ pt \sqrt &= \frac \end


Table

These can be shown by using either the sum and difference identities or the multiple-angle formulae. The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a
compass and straightedge construction In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideal ...
of
angle trisection Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge an ...
to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory. A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation , where x is the value of the cosine function at the one-third angle and is the known value of the cosine function at the full angle. However, the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions is reducible to a real
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
, as they use intermediate complex numbers under the
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. F ...
s.


Power-reduction formulae

Obtained by solving the second and third versions of the cosine double-angle formula. In general terms of powers of \sin \theta or \cos \theta the following is true, and can be deduced using De Moivre's formula,
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
and the binomial theorem .


Product-to-sum and sum-to-product identities

The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. See
amplitude modulation Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting messages with a radio wave. In amplitude modulation, the amplitude (signal strength) of the wave is varied in proportion to ...
for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.


Hermite's cotangent identity

Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ...
demonstrated the following identity. Suppose a_1, \ldots, a_n are
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, no two of which differ by an integer multiple of . Let :A_ = \prod_ \cot(a_k - a_j) (in particular, A_, being an empty product, is 1). Then :\cot(z - a_1)\cdots\cot(z - a_n) = \cos\frac + \sum_^n A_ \cot(z - a_k). The simplest non-trivial example is the case : :\cot(z - a_1)\cot(z - a_2) = -1 + \cot(a_1 - a_2)\cot(z - a_1) + \cot(a_2 - a_1)\cot(z - a_2).


Finite products of trigonometric functions

For
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
integers , :\prod_^n \left(2a + 2\cos\left(\frac + x\right)\right) = 2\left( T_n(a)+^\cos(n x) \right) where is the
Chebyshev polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebys ...
. The following relationship holds for the sine function :\prod_^ \sin\left(\frac\right) = \frac. More generally for an integer :\sin(nx) = 2^\prod_^ \sin\left(\frac\pi + x\right) = 2^\prod_^ \sin\left(\frac\pi - x\right). or written in terms of the chord function \operatornamex \equiv 2\sin\tfrac12x, :\operatorname(nx) = \prod_^ \operatorname\left(\frac2\pi - x\right). This comes from the factorization of the polynomial z^n - 1 into linear factors (cf.
root of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
): For a point on the complex unit circle and an integer , :z^n - 1 = \prod_^ z - \exp\Bigl(\frac2\pi i\Bigr).


Linear combinations

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different
phase shifts In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
is also a sine wave with the same period or frequency, but a different phase shift. This is useful in
sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
data fitting, because the measured or observed data are linearly related to the and unknowns of the in-phase and quadrature components basis below, resulting in a simpler
Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: * Jacobian matrix and determinant * Jacobian elliptic functions * Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähle ...
, compared to that of c and \varphi.


Sine and cosine

The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude, :a\cos x+b\sin x=c\cos(x+\varphi) where c and \varphi are defined as so: :\begin c &= \sgn(a) \sqrt, \\ \varphi &= \arctan \left(-\frac\right), \end given that a \neq 0.


Arbitrary phase shift

More generally, for arbitrary phase shifts, we have :a \sin(x + \theta_a) + b \sin(x + \theta_b)= c \sin(x+\varphi) where c and \varphi satisfy: :\begin c^2 &= a^2 + b^2 + 2ab\cos \left(\theta_a - \theta_b \right) , \\ \tan \varphi &= \frac. \end


More than two sinusoids

The general case reads :\sum_i a_i \sin(x + \theta_i) = a \sin(x + \theta), where :a^2 = \sum_a_i a_j \cos(\theta_i - \theta_j) and :\tan\theta = \frac.


Lagrange's trigonometric identities

These identities, named after Joseph Louis Lagrange, are: \begin \sum_^n \sin k\theta & = \frac\\ pt\sum_^n \cos k\theta & = \frac \end for \theta \not\equiv 0 \pmod. A related function is the Dirichlet kernel: D_n(\theta) = 1 + 2\sum_^n \cos k\theta = \frac.


Certain linear fractional transformations

If f(x) is given by the
linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
f(x) = \frac, and similarly g(x) = \frac, then f\big(g(x)\big) = g\big(f(x)\big) = \frac. More tersely stated, if for all \alpha we let f_ be what we called f above, then f_\alpha \circ f_\beta = f_. If x is the slope of a line, then f(x) is the slope of its rotation through an angle of - \alpha.


Relation to the complex exponential function

Euler's formula states that, for any real number ''x'': e^ = \cos x + i\sin x, where ''i'' is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. Substituting −''x'' for ''x'' gives us: e^ = \cos(-x) + i\sin(-x) = \cos x - i\sin x. These two equations can be used to solve for cosine and sine in terms of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
. Specifically, \cos x = \frac \sin x = \frac These formulae are useful for proving many other trigonometric identities. For example, that means that That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine. The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the
complex logarithm In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to b ...
.


Infinite product formulae

For applications to
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined b ...
, the following
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
formulae for trigonometric functions are useful: \begin \sin x &= x \prod_^\infty\left(1 - \frac\right) & \cos x &= \prod_^\infty\left(1 - \frac\right) \\ \sinh x &= x \prod_^\infty\left(1 + \frac\right) & \cosh x &= \prod_^\infty\left(1 + \frac\right) \end


Inverse trigonometric functions

The following identities give the result of composing a trigonometric function with an inverse trigonometric function. \begin \sin(\arcsin x) &=x & \cos(\arcsin x) &=\sqrt & \tan(\arcsin x) &=\frac \\ \sin(\arccos x) &=\sqrt & \cos(\arccos x) &=x & \tan(\arccos x) &=\frac \\ \sin(\arctan x) &=\frac & \cos(\arctan x) &=\frac & \tan(\arctan x) &=x \\ \sin(\arccsc x) &=\frac & \cos(\arccsc x) &=\frac & \tan(\arccsc x) &=\frac \\ \sin(\arcsec x) &=\frac & \cos(\arcsec x) &=\frac & \tan(\arcsec x) &=\sqrt \\ \sin(\arccot x) &=\frac & \cos(\arccot x) &=\frac & \tan(\arccot x) &=\frac \\ \end Taking the
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
of both sides of the each equation above results in the equations for \csc = \frac, \;\sec = \frac, \text \cot = \frac. The right hand side of the formula above will always be flipped. For example, the equation for \cot(\arcsin x) is: \cot(\arcsin x) = \frac = \frac = \frac while the equations for \csc(\arccos x) and \sec(\arccos x) are: \csc(\arccos x) = \frac = \frac \qquad \text\quad \sec(\arccos x) = \frac = \frac. The following identities are implied by the reflection identities. They hold whenever x, r, s, -x, -r, \text -s are in the domains of the relevant functions. \begin \frac ~&=~ \arcsin(x) &&+ \arccos(x) ~&&=~ \arctan(r) &&+ \arccot(r) ~&&=~ \arcsec(s) &&+ \arccsc(s) \\ .4ex\pi ~&=~ \arccos(x) &&+ \arccos(-x) ~&&=~ \arccot(r) &&+ \arccot(-r) ~&&=~ \arcsec(s) &&+ \arcsec(-s) \\ .4ex0 ~&=~ \arcsin(x) &&+ \arcsin(-x) ~&&=~ \arctan(r) &&+ \arctan(-r) ~&&=~ \arccsc(s) &&+ \arccsc(-s) \\ .0ex\end Also,Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", ''Mathematics Magazine'' 77(3), June 2004, p. 189. \begin \arctan x + \arctan \dfrac &= \begin \frac, & \text x > 0 \\ - \frac, & \text x < 0 \end \\ \arccot x + \arccot \dfrac &= \begin \frac, & \text x > 0 \\ \frac, & \text x < 0 \end \\ \end \arccos \frac = \arcsec x \qquad \text \qquad \arcsec \frac = \arccos x \arcsin \frac = \arccsc x \qquad \text \qquad \arccsc \frac = \arcsin x The
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
function can be expanded as a series: \arctan(nx) = \sum_^n \arctan\frac


Identities without variables

In terms of the
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
function we have \arctan \frac = \arctan \frac + \arctan \frac. The curious identity known as Morrie's law, \cos 20^\circ\cdot\cos 40^\circ\cdot\cos 80^\circ = \frac, is a special case of an identity that contains one variable: \prod_^\cos\left(2^j x\right) = \frac. Similarly, \sin 20^\circ\cdot\sin 40^\circ\cdot\sin 80^\circ = \frac is a special case of an identity with x = 20^\circ: \sin x \cdot \sin \left(60^\circ - x\right) \cdot \sin \left(60^\circ + x\right) = \frac. For the case x = 15^\circ, \begin \sin 15^\circ\cdot\sin 45^\circ\cdot\sin 75^\circ &= \frac, \\ \sin 15^\circ\cdot\sin 75^\circ &= \frac. \end For the case x = 10^\circ, \sin 10^\circ\cdot\sin 50^\circ\cdot\sin 70^\circ = \frac. The same cosine identity is \cos x \cdot \cos \left(60^\circ - x\right) \cdot \cos \left(60^\circ + x\right) = \frac. Similarly, \begin \cos 10^\circ\cdot\cos 50^\circ\cdot\cos 70^\circ &= \frac, \\ \cos 15^\circ\cdot\cos 45^\circ\cdot\cos 75^\circ &= \frac, \\ \cos 15^\circ\cdot\cos 75^\circ &= \frac. \end Similarly, \begin \tan 50^\circ\cdot\tan 60^\circ\cdot\tan 70^\circ &= \tan 80^\circ, \\ \tan 40^\circ\cdot\tan 30^\circ\cdot\tan 20^\circ &= \tan 10^\circ. \end The following is perhaps not as readily generalized to an identity containing variables (but see explanation below): \cos 24^\circ + \cos 48^\circ + \cos 96^\circ + \cos 168^\circ = \frac. Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators: \cos \frac + \cos\left(2\cdot\frac\right) + \cos\left(4\cdot\frac\right) + \cos\left( 5\cdot\frac\right) + \cos\left( 8\cdot\frac\right) + \cos\left(10\cdot\frac\right) = \frac. The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than that are relatively prime to (or have no
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s in common with) 21. The last several examples are corollaries of a basic fact about the irreducible
cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitiv ...
s: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively. Other cosine identities include: \begin 2\cos \frac &= 1, \\ 2\cos \frac \times 2\cos \frac &= 1, \\ 2\cos \frac \times 2\cos \frac\times 2\cos \frac &= 1, \end and so forth for all odd numbers, and hence \cos \frac+\cos \frac \times \cos \frac + \cos \frac \times \cos \frac \times \cos \frac + \dots = 1. Many of those curious identities stem from more general facts like the following: \prod_^ \sin\frac = \frac and \prod_^ \cos\frac = \frac. Combining these gives us \prod_^ \tan\frac = \frac If is an odd number (n = 2 m + 1) we can make use of the symmetries to get \prod_^ \tan\frac = \sqrt The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved: \prod_^n \sin\frac = \prod_^ \cos\frac = \frac


Computing

An efficient way to compute to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula: \frac = 4 \arctan\frac - \arctan\frac or, alternatively, by using an identity of
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
: \frac = 5 \arctan\frac + 2 \arctan\frac or by using Pythagorean triples: \pi = \arccos\frac + \arccos\frac + \arccos\frac = \arcsin\frac + \arcsin\frac + \arcsin\frac. Others include:Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, ''Proofs Without Words'' (1993, Mathematical Association of America), p. 39. \frac = \arctan\frac + \arctan\frac, \pi = \arctan 1 + \arctan 2 + \arctan 3, \frac = 2\arctan \frac + \arctan \frac. Generally, for numbers for which , let . This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are and its value will be in . In particular, the computed will be rational whenever all the values are rational. With these values, \begin \frac & = \sum_^n \arctan(t_k) \\ \pi & = \sum_^n \sgn(t_k) \arccos\left(\frac\right) \\ \pi & = \sum_^n \arcsin\left(\frac\right) \\ \pi & = \sum_^n \arctan\left(\frac\right)\,, \end where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the values is not within . Note that if is rational, then the values in the above formulae are proportional to the Pythagorean triple . For example, for terms, \frac = \arctan\left(\frac\right) + \arctan\left(\frac\right) + \arctan\left(\frac\right) for any .


An identity of Euclid

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
showed in Book XIII, Proposition 10 of his '' Elements'' that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says: \sin^2 18^\circ + \sin^2 30^\circ = \sin^2 36^\circ.
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
used this proposition to compute some angles in his table of chords in Book I, chapter 11 of ''
Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it can ...
''.


Composition of trigonometric functions

These identities involve a trigonometric function of a trigonometric function: :\cos(t \sin x) = J_0(t) + 2 \sum_^\infty J_(t) \cos(2kx) :\sin(t \sin x) = 2 \sum_^\infty J_(t) \sin\big((2k+1)x\big) :\cos(t \cos x) = J_0(t) + 2 \sum_^\infty (-1)^kJ_(t) \cos(2kx) :\sin(t \cos x) = 2 \sum_^\infty(-1)^k J_(t) \cos\big((2k+1)x\big) where are
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrar ...
s.


Further "conditional" identities for the case ''α'' + ''β'' + ''γ'' = 180°

The following formulae apply to arbitrary plane triangles and follow from \alpha + \beta + \gamma = 180^, as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur). \begin \tan \alpha + \tan \beta + \tan \gamma &= \tan \alpha \tan \beta \tan \gamma \\ 1 &= \cot \beta \cot \gamma + \cot \gamma \cot \alpha + \cot \alpha \cot \beta \\ \cot\left(\frac\right) + \cot\left(\frac\right) + \cot\left(\frac\right) &= \cot\left(\frac\right) \cot \left(\frac\right) \cot\left(\frac\right) \\ 1 &= \tan\left(\frac\right)\tan\left(\frac\right) + \tan\left(\frac\right)\tan\left(\frac\right) + \tan\left(\frac\right)\tan\left(\frac\right) \\ \sin \alpha + \sin \beta + \sin \gamma &= 4\cos\left(\frac\right)\cos\left(\frac\right)\cos\left(\frac\right) \\ -\sin \alpha + \sin \beta + \sin \gamma &= 4\cos\left(\frac\right)\sin\left(\frac\right)\sin\left(\frac\right) \\ \cos \alpha + \cos \beta + \cos \gamma &= 4\sin\left(\frac\right)\sin\left(\frac\right)\sin \left(\frac\right) + 1 \\ -\cos \alpha + \cos \beta + \cos \gamma &= 4\sin\left(\frac\right)\cos\left(\frac\right)\cos \left(\frac\right) - 1 \\ \sin (2\alpha) + \sin (2\beta) + \sin (2\gamma) &= 4\sin \alpha \sin \beta \sin \gamma \\ -\sin (2\alpha) + \sin (2\beta) + \sin (2\gamma) &= 4\sin \alpha \cos \beta \cos \gamma \\ \cos (2\alpha) + \cos (2\beta) + \cos (2\gamma) &= -4\cos \alpha \cos \beta \cos \gamma - 1 \\ -\cos (2\alpha) + \cos (2\beta) + \cos (2\gamma) &= -4\cos \alpha \sin \beta \sin \gamma + 1 \\ \sin^2\alpha + \sin^2\beta + \sin^2\gamma &= 2 \cos \alpha \cos \beta \cos \gamma + 2 \\ -\sin^2\alpha + \sin^2\beta + \sin^2\gamma &= 2 \cos \alpha \sin \beta \sin \gamma \\ \cos^2\alpha + \cos^2\beta + \cos^2\gamma &= -2 \cos \alpha \cos \beta \cos \gamma + 1 \\ -\cos^2\alpha + \cos^2\beta + \cos^2\gamma &= -2 \cos \alpha \sin \beta \sin \gamma + 1 \\ \sin^2 (2\alpha) + \sin^2 (2\beta) + \sin^2 (2\gamma) &= -2\cos (2\alpha) \cos (2\beta) \cos (2\gamma)+2 \\ \cos^2 (2\alpha) + \cos^2 (2\beta) + \cos^2 (2\gamma) &= 2\cos (2\alpha) \,\cos (2\beta) \,\cos (2\gamma) + 1 \\ 1 &= \sin^2 \left(\frac\right) + \sin^2 \left(\frac\right) + \sin^2 \left(\frac\right) + 2\sin \left(\frac\right) \,\sin \left(\frac\right) \,\sin \left(\frac\right) \end


Historical shorthands

The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.


Miscellaneous


Relationship between all trigonometric ratios

The following identities each give a relationship between all the trigonometric ratios. :(\sin\theta + \csc\theta)^2 + (\cos\theta + \sec\theta)^2 - (\tan\theta + \cot\theta)^2 = 5 :(\sin\theta + \csc\theta)^2 + (\cos\theta + \sec\theta)^2 - (\tan\theta - \cot\theta)^2 = 9 Similarly, :(\sin\theta + \csc\theta)^2 + (\cos\theta + \sec\theta)^2 = \tan^2\theta + \cot^2\theta + 7


Dirichlet kernel

The Dirichlet kernel is the function occurring on both sides of the next identity: 1 + 2\cos x + 2\cos(2x) + 2\cos(3x) + \cdots + 2\cos(nx) = \frac. The
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of any integrable function of period 2 \pi with the Dirichlet kernel coincides with the function's nth-degree Fourier approximation. The same holds for any measure or generalized function.


Tangent half-angle substitution

If we set t = \tan\frac x 2, thenAbramowitz and Stegun, p. 72, 4.3.23 \sin x = \frac;\qquad \cos x = \frac;\qquad e^ = \frac where e^ = \cos x + i \sin x, sometimes abbreviated to . When this substitution of t for is used in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, it follows that \sin x is replaced by , \cos x is replaced by and the differential is replaced by . Thereby one converts rational functions of \sin x and \cos x to rational functions of t in order to find their
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s.


Viète's infinite product

\cos\frac \cdot \cos \frac \cdot \cos \frac \cdots = \prod_^\infty \cos \frac = \frac = \operatorname \theta.


See also

* Aristarchus's inequality * Derivatives of trigonometric functions * Exact trigonometric values (values of sine and cosine expressed in surds) * Exsecant *
Half-side formula In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles. F ...
*
Hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
* Laws for solution of triangles: ** Law of cosines *** Spherical law of cosines ** Law of sines **
Law of tangents In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , ...
**
Law of cotangents In trigonometry, the law of cotangentsThe Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960. is a relationship am ...
**
Mollweide's formula In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. A variant in more geometrical style was first published by Isaac Newton in 1707 and then by in 1746. Thomas Simpson published the now-stand ...
* List of integrals of trigonometric functions * Mnemonics in trigonometry * Pentagramma mirificum *
Proofs of trigonometric identities There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of ...
* Prosthaphaeresis *
Pythagorean theorem 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 (the side opposit ...
*
Tangent half-angle formula 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 an angle is the stereographic projection of the circle onto a line. Among these formulas are the ...
* Trigonometric number *
Trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
* Trigonometric constants expressed in real radicals *
Uses of trigonometry Amongst the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still ...
* Versine and haversine


References


Bibliography

* * *


External links


Values of sin and cos, expressed in surds, for integer multiples of 3° and of °
and for the same angle

an


Complete List of Trigonometric Formulas
{{DEFAULTSORT:Trigonometric identities Mathematical identities Identities Mathematics-related lists