List of trigonometric identities


In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
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.

Notation

Angles

This article uses Greek letters such as alpha, beta, gamma, and theta to represent angles. Several different units of angle measure are widely used, including degree, radian, and gradian :
If not specifically annotated by for degree or for gradian, all values for angles in this article are assumed to be given in radian.
The following table shows for some common angles their conversions and the values of the basic trigonometric functions:
TurnDegreeRadianGradiansinecosinetangent
Undefined
Undefined

Results for other angles can be found at Trigonometric constants expressed in real radicals. Per Niven's theorem, are the only rational numbers that, taken in degrees, result in a rational sine-value for the corresponding angle within the first turn, which may account for their popularity in examples. The analogous condition for the unit radian requires that the argument divided by is rational, and yields the solutions 0, /6, /2, 5/6,, 7/6, 3/2, 11/6.

Trigonometric functions

The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. Their usual abbreviations are, and, respectively, where denotes the angle. The parentheses around the argument of the functions are often omitted, e.g., and, if an interpretation is unambiguously possible.
The sine of an angle is defined, in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle.
The cosine of an angle in this context is the ratio of the length of the side that is adjacent to the angle divided by the length of the hypotenuse.
The tangent of an angle in this context is the ratio of the length of the side that is opposite to the angle divided by the length of the side that is adjacent to the angle. This is the same as the ratio of the sine to the cosine of this angle, as can be seen by substituting the definitions of and from above:
The remaining trigonometric functions secant, cosecant, and cotangent are defined as the reciprocal functions of cosine, sine, and tangent, respectively. Rarely, these are called the secondary trigonometric functions:
These definitions are sometimes referred to as ratio identities.

Inverse functions

The inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine or arcsine, satisfies
and
This article uses the notation below for inverse trigonometric functions:
Functionsincostanseccsccot
Inversearcsinarccosarctanarcsecarccscarccot

Pythagorean identities

In trigonometry, the basic relationship between the sine and the cosine is given by the Pythagorean identity:
where means and means.
This can be viewed as a version of the Pythagorean theorem, and follows from the equation for the unit circle. This equation can be solved for either the sine or the cosine:
where the sign depends on the quadrant of.
Dividing this identity by either or yields the other two Pythagorean identities:
Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other :
in terms of

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.
NameAbbreviationValue
versed sine, versine

versed cosine, vercosine

coversed sine, coversine

coversed cosine, covercosine

half versed sine, haversine

half versed cosine, havercosine

half coversed sine, hacoversine
cohaversine

half coversed cosine, hacovercosine
cohavercosine

exterior secant, exsecant
exterior cosecant, excosecant

chord

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, this is the angle determined by the free vector and the positive -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 -axis. If a line with direction is reflected about a line with direction then the direction angle of this reflected line has the value
The values of the trigonometric functions of these angles for specific angles satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as reduction formulae.
reflected in = 0
odd/even identities		 reflected in = reflected in = reflected in =
compare to = 0

Shifts and periodicity

Through shifting the arguments of trigonometric functions by certain angles, changing the sign or applying complementary trigonometric functions can sometimes express particular results more simply. Some examples of shifts are shown below in the table.
Shift by one quarter periodShift by one half periodShift by full periodsPeriod

Angle sum and difference identities

These are also known as the angle addition and subtraction theorems.
The identities can be derived by combining right triangles such as in the adjacent diagram, or by considering the invariance of the length of a chord on a unit circle given a particular central angle. The most intuitive derivation uses rotation matrices.
For acute angles and, whose sum is non-obtuse, a concise diagram illustrates the angle sum formulae for sine and cosine: The bold segment labeled "1" has unit length and serves as the hypotenuse of a right triangle with angle ; the opposite and adjacent legs for this angle have respective lengths and. The leg is itself the hypotenuse of a right triangle with angle ; that triangle's legs, therefore, have lengths given by and, multiplied by. The leg, as hypotenuse of another right triangle with angle, likewise leads to segments of length and. Now, we observe that the "1" segment is also the hypotenuse of a right triangle with angle ; the leg opposite this angle necessarily has length, while the leg adjacent has length. Consequently, as the opposing sides of the diagram's outer rectangle are equal, we deduce
Relocating one of the named angles yields a variant of the diagram that demonstrates the angle difference formulae for sine and cosine. Dividing all elements of the diagram by provides yet another variant illustrating the angle sum formula for tangent.
These identities have applications in, for example, in-phase and quadrature components.
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
Arcsine
Arccosine
Arctangent
Arccotangent

Matrix form

The sum and difference formulae for sine and cosine follow from the fact that a rotation of the plane by angle α, following a rotation by β, is equal to a rotation by α+β. In terms of rotation matrices:
The matrix inverse for a rotation is the rotation with the negative of the angle
which is also the matrix transpose.
These formulae show that these matrices form a representation of the rotation group in the plane, since the composition law is fulfilled and inverses exist. Furthermore, matrix multiplication of the rotation matrix for an angle with a column vector will rotate the column vector counterclockwise by the angle.
Since multiplication by a complex number of unit length rotates the complex plane by the argument of the number, the above multiplication of rotation matrices is equivalent to a multiplication of complex numbers:
In terms of Euler's formula, this simply says, showing that is a one-dimensional complex representation of.

Sines and cosines of sums of infinitely many angles

When the series converges absolutely then
Because the series converges absolutely, it is necessarily the case that,, and. 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 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 be the th-degree elementary symmetric polynomial in the variables
for = 0, 1, 2, 3, ..., i.e.,
Then
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:
and so on. The case of only finitely many terms can be proved by mathematical induction.

Secants and cosecants of sums

where is the th-degree elementary symmetric polynomial in the variables,, 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,

Multiple-angle formulae

Double-angle, triple-angle, and half-angle formulae

Double-angle formulae

Formulae for twice an angle.

Triple-angle formulae

Formulae for triple angles.

Half-angle formulae

Also

Table

These can be shown by using either the sum and difference identities or the multiple-angle formulae.
SineCosineTangentCotangent
Double-angle formulae
Triple-angle formulae
Half-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 of angle trisection 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 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. None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Sine, cosine, and tangent of multiple angles

For specific multiples, these follow from the angle addition formulae, while the general formula was given by 16th-century French mathematician François Viète.
for nonnegative values of up through.
In each of these two equations, the first parenthesized term is a binomial coefficient, and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed. The ratio of these formulae gives

Chebyshev method

The Chebyshev method is a recursive algorithm for finding the th multiple angle formula knowing the th and th values.
can be computed from,, and with
This can be proved by adding together the formulae
It follows by induction that is a polynomial of, the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.
Similarly, can be computed from,, and with
This can be proved by adding formulae for and.
Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

Tangent of an average

Setting either or to 0 gives the usual tangent half-angle formulae.

Viète's infinite product

Power-reduction formulae

Obtained by solving the second and third versions of the cosine double-angle formula.
SineCosineOther

and in general terms of powers of or the following is true, and can be deduced using De Moivre's formula, Euler's formula 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. See amplitude modulation for an application of the product-to-sum formulae, and beat and phase detector for applications of the sum-to-product formulae.
Product-to-sum

Other related identities

demonstrated the following identity. Suppose are complex numbers, no two of which differ by an integer multiple of . Let
. Then
The simplest non-trivial example is the case :

Ptolemy's theorem

Ptolemy's theorem can be expressed in the language of modern trigonometry as:

Finite products of trigonometric functions

For coprime integers,
where is the Chebyshev polynomial.
The following relationship holds for the sine function
More generally

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 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 basis below, resulting in a simpler Jacobian, compared to that of and.

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,
where and are defined as so:

Arbitrary phase shift

More generally, for arbitrary phase shifts, we have
where and satisfy:

More than two sinusoids

The general case reads
where
and
See also Phasor addition.

Lagrange's trigonometric identities

These identities, named after Joseph Louis Lagrange, are:
A related function is the following function of, called the Dirichlet kernel.
see proof.

Other sums of trigonometric functions

Sum of sines and cosines with arguments in arithmetic progression: if, then
The above identity is sometimes convenient to know when thinking about the Gudermannian function, which relates the circular and hyperbolic trigonometric functions without resorting to complex numbers.
If,, and are the three angles of any triangle, i.e. if, then

Certain linear fractional transformations

If is given by the linear fractional transformation
and similarly
then
More tersely stated, if for all we let be what we called above, then
If is the slope of a line, then is the slope of its rotation through an angle of.

Inverse trigonometric functions

Compositions of trig and inverse trig functions

Relation to the complex exponential function

With the unit imaginary number satisfying,
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.

Infinite product formulae

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:

Identities without variables

In terms of the arctangent function we have
The curious identity known as Morrie's law,
is a special case of an identity that contains one variable:
The same cosine identity in radians is
Similarly,
is a special case of an identity with the case x = 20:
For the case = 15,
For the case = 10,
The same cosine identity is
Similarly,
Similarly,
The following is perhaps not as readily generalized to an identity containing variables :
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
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 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at 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:
and so forth for all odd numbers, and hence
Many of those curious identities stem from more general facts like the following:
and
Combining these gives us
If is an odd number we can make use of the symmetries to get
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:

Computing

An efficient way to compute is based on the following identity without variables, due to Machin:
or, alternatively, by using an identity of Leonhard Euler:
or by using Pythagorean triples:
Others include
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,
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 when is rational then the values in the above formulae are proportional to the Pythagorean triple.
For example, for terms,
for any.

A useful mnemonic for certain values of sines and cosines

For certain simple angles, the sines and cosines take the form for, which makes them easy to remember.

Miscellany

With the golden ratio :
Also see trigonometric constants expressed in real radicals.

An identity of Euclid

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:
Ptolemy used this proposition to compute some angles in his table of chords.

Composition of trigonometric functions

This identity involves a trigonometric function of a trigonometric function:
where are Bessel functions.

Calculus

In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, along with the definitions of arc length and area, their derivatives can be found by verifying two limits. The first is:
verified using the unit circle and squeeze theorem. The second limit is:
verified using the identity. Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that and. If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.
The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation:
The integral identities can be found in List of integrals of trigonometric functions. Some generic forms are listed below.

Implications

The fact that the differentiation of trigonometric functions results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and Fourier transforms.

Some differential equations satisfied by the sine function

Let be the imaginary unit and let ∘ denote composition of differential operators. Then for every odd positive integer ,
This identity was discovered as a by-product of research in medical imaging.

Exponential definitions

FunctionInverse function
cis |

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

The following formulae apply to arbitrary plane triangles and follow from α + β + γ = 180°, as long as the functions occurring in the formulae are well-defined.

Miscellaneous

Dirichlet kernel

The Dirichlet kernel is the function occurring on both sides of the next identity:
The convolution of any integrable function of period 2 with the Dirichlet kernel coincides with the function's th-degree Fourier approximation. The same holds for any measure or generalized function.

Tangent half-angle substitution

If we set
then
where, sometimes abbreviated to .
When this substitution of for is used in calculus, it follows that is replaced by, is replaced by and the differential is replaced by. Thereby one converts rational functions of and to rational functions of in order to find their antiderivatives.