List Of Trigonometric Identities
   HOME

TheInfoList



OR:

In
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 al ...
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 a ...
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 Edge (geometry), edges and three Vertex (geometry), 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, an ...
. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
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 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 oppo ...
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 opposite t ...
, 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 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, \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 In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
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 mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
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 mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
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 Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions be ...
). 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 In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
, \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 Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
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 ideali ...
of angle trisection to the algebraic problem of solving a
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
, 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 In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
, 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 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 roots.


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 In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that :\big(\cos x + i \sin x\big)^n = \cos nx + i \sin nx, where is the imaginary unit (). ...
,
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 fo ...
and the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
.


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

The product-to-sum identities or
prosthaphaeresis Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the ...
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 Johann(es) Werner ( la, Ioannes Vernerus; February 14, 1468 – May 1522) was a German mathematician. He was born in Nuremberg, Germany, where he became a parish priest. His primary work was in astronomy, mathematics, and geography, although he ...
who used them for astronomical calculations. See amplitude modulation for an application of the product-to-sum formulae, and
beat (acoustics) In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, ''perceived'' as a periodic variation in volume whose rate is the difference of the two frequencies. With tuning instruments that can produce ...
and
phase detector A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a signal which represents the difference in phase between two signal inputs. The phase detector is an essential element of the phase- ...
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. Hermi ...
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 form ...
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 In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
, 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 Chebyshe ...
. 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 ...
): 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 data fitting, because the measured or observed data are linearly related to the and unknowns of the
in-phase and quadrature components In electrical engineering, a sinusoid with angle modulation can be decomposed into, or synthesized from, two amplitude-modulated sinusoids that are offset in phase by one-quarter cycle (90 degrees or /2 radians). All three functions have the s ...
basis below, resulting in a simpler Jacobian, 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 Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia\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 In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonneg ...
: 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 transf ...
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.


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 function can be expanded as a series: \arctan(nx) = \sum_^n \arctan\frac


Identities without variables

In terms of the arctangent function we have \arctan \frac = \arctan \frac + \arctan \frac. The curious identity known as
Morrie's law Morrie's law is a special trigonometric identity. Its name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morr ...
, \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 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 ...
to (or have no prime factors 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 primiti ...
s: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
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 In mathematics, Machin-like formulae are a popular technique for computing to a large number of digits. They are generalizations of John Machin's formula from 1706: :\frac = 4 \arctan \frac - \arctan \frac which he used to compute to 100 de ...
: \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 in ma ...
: \frac = 5 \arctan\frac + 2 \arctan\frac or by using
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s: \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 importance ...
used this proposition to compute some angles in his table of chords in Book I, chapter 11 of '' Almagest''.


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 The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',coversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',haversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',exsecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astro ...
were used in navigation. For example, the
haversine formula 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, ...
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 In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonneg ...
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 In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
of period 2 \pi with the Dirichlet kernel coincides with the function's nth-degree Fourier approximation. The same holds for any
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
or
generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
.


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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, 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 symbolicall ...
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 Aristarchus's inequality (after the Greek astronomer and mathematician Aristarchus of Samos; c. 310 – c. 230 BCE) is a law of trigonometry which states that if ''α'' and ''β'' are acute angles (i.e. between 0 and a right angle) an ...
* Derivatives of trigonometric functions *
Exact trigonometric values In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values ...
(values of sine and cosine expressed in surds) *
Exsecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astro ...
*
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. Fo ...
*
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 In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
***
Spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "sph ...
**
Law of sines 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, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and ar ...
**
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-standar ...
*
List of integrals of trigonometric functions The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antid ...
*
Mnemonics in trigonometry In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions. SOH-CAH-TOA The ''sine'', ''cosine'', and ''tangent'' ratios in a right triangle can be ...
*
Pentagramma mirificum Pentagramma mirificum (Latin for ''miraculous pentagram'') is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book '' Mirifici Lo ...
*
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 r ...
*
Prosthaphaeresis Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding 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 opposite t ...
*
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 th ...
*
Trigonometric number In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values ...
*
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 In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values f ...
*
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 music theory, theory of ...
*
Versine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',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