List of trigonometric identities

List of trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions 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 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.
Pythagorean identities
The basic relationship between the sine and 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 or both yields the following identities:
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 this is the angle determined by the free vector (starting at the origin) 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 (vector) with direction is reflected about a line with direction then the direction angle of this reflected line (vector) 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 .
Shifts and periodicity
Signs
The sign of trigonometric functions depends on quadrant of the angle. If and is the sign function,
The trigonometric functions are periodic with common period so for values of outside the interval they take repeating values (see above).
Angle sum and difference identities
These are also known as the (or ).
The angle difference identities for and can be derived from the angle sum versions by substituting for and using the facts that and . 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 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 (for ) be the th-degree elementary symmetric polynomial in the variables
for that is,
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,
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. It states that in a cyclic quadrilateral , 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, and are both right angles. The right-angled triangles and both share the hypotenuse of length 1. Thus, the side and .
By the inscribed angle theorem, the central angle subtended by the chord at the circle's center is twice the angle , i.e. . Therefore, the symmetrical pair of red triangles each has the angle at the center. Each of these triangles has a hypotenuse of length , so the length of is , i.e. simply . The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also .
When these values are substituted into the statement of Ptolemy's theorem that , this yields the angle sum trigonometric identity for sine: . The angle difference formula for can be similarly derived by letting the side serve as a diameter instead of .
Multiple-angle and half-angle formulae
Multiple-angle formulae
Double-angle formulae
Formulae for twice an angle.
Triple-angle formulae
Formulae for triple angles.
Multiple-angle formulae
Formula for multiple angles.
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:
Half-angle formulae
Also
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 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 (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, 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 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. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.
Product-to-sum identities
Sum-to-product identities
The sum-to-product identities are as follows:
Hermite's cotangent identity
Charles Hermite demonstrated the following identity. Suppose are complex numbers, no two of which differ by an integer multiple of . Let
(in particular, being an empty product, is 1). Then
The simplest non-trivial example is the case :
Finite products of trigonometric functions
For coprime integers ,
where is the Chebyshev polynomial.
The following relationship holds for the sine function
More generally for an integer
or written in terms of the chord function ,
This comes from the factorization of the polynomial into linear factors (cf. root of unity): For a point on the complex unit circle and an integer ,
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:
given that
Arbitrary phase shift
More generally, for arbitrary phase shifts, we have
where and satisfy:
More than two sinusoids
The general case reads
where
and
Lagrange's trigonometric identities
These identities, named after Joseph Louis Lagrange, are:
for
A related function is the Dirichlet kernel:
A similar identity is
The proof is the following. By using the angle sum and difference identities,
Then let's examine the following formula,
and this formula can be written by using the above identity,
So, dividing this formula with completes the proof.
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
Relation to the complex exponential function
Euler's formula states that, for any real number x:
where i is the imaginary unit. Substituting −x for x gives us:
These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically,
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, the following infinite product formulae for trigonometric functions are useful:
Infinite sums
Using the identity , and substituting in , one can derivefor . Both sides being periodic with period . Integrating with respect to yieldsand so on.
Inverse trigonometric functions
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.
Taking the multiplicative inverse of both sides of the each equation above results in the equations for
The right hand side of the formula above will always be flipped.
For example, the equation for is:
while the equations for and are:
The following identities are implied by the reflection identities. They hold whenever are in the domains of the relevant functions.
Also,
The arctangent function can be expanded as a series:
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:
Similarly,
is a special case of an identity with :
For the case ,
For the case ,
The same cosine identity is
Similarly,
Similarly,
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
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 (or have no prime factors in common with) 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 (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:
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 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:
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 if is rational, then the values in the above formulae are proportional to the Pythagorean triple .
For example, for terms,
for any .
An identity of Euclid
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 in Book I, chapter 11 of Almagest.
Composition of trigonometric functions
These identities involve a trigonometric function of a trigonometric function:
where are Bessel functions.
Further "conditional" identities for the case α + β + γ = 180°
The following formulae apply to arbitrary plane triangles and follow from 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).
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.
Similarly,
Dirichlet kernel
The Dirichlet kernel is the function occurring on both sides of the next identity:
The convolution of any integrable function of period 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.
Viète's infinite product
See also
Aristarchus's inequality
Derivatives of trigonometric functions
Exact trigonometric values (values of sine and cosine expressed in surds)
Exsecant
Half-side formula
Hyperbolic function
Laws for solution of triangles:
Law of cosines
Spherical law of cosines
Law of sines
Law of tangents
Law of cotangents
Mollweide's formula
List of integrals of trigonometric functions
Mnemonics in trigonometry
Pentagramma mirificum
Proofs of trigonometric identities
Prosthaphaeresis
Pythagorean theorem
Tangent half-angle formula
Trigonometric number
Trigonometry
Uses of trigonometry
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 angles csc and sec and tan
Complete List of Trigonometric Formulas
Mathematical identities
Identities
Mathematics-related lists