List of trigonometric identities

Summary

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 identitiesEdit

 
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity  , and the red triangle shows that  .

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):

Each trigonometric function in terms of each of the other five.[1]
in terms of            
             
             
             
   
         
             
             

Reflections, shifts, and periodicityEdit

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

ReflectionsEdit

 
Transformation of coordinates (a,b) when shifting the reflection angle   in increments of  .

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 reduction formulae.[2]

  reflected in  [3]
odd/even identities
  reflected in     reflected in     reflected in     reflected in  
compare to  
         
         
         
         
         
         

Shifts and periodicityEdit

 
Transformation of coordinates (a,b) when shifting the angle   in increments of  .
Shift by one quarter period Shift by one half period Shift by full periods[4] Period
       
       
       
       
       
       

Angle sum and difference identitiesEdit

 
Illustration of angle addition formulae for the sine and cosine of acute angles. Emphasized segment is of unit length.

These are also known as the angle addition and subtraction theorems (or formulae).

 

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.

Sine      [5][6]
Cosine      [6][7]
Tangent      [6][8]
Cosecant      [9]
Secant      [9]
Cotangent      [6][10]
Arcsine      [11]
Arccosine      [12]
Arctangent      [13]
Arccotangent      

Sines and cosines of sums of infinitely many anglesEdit

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 sumsEdit

Let   (for  ) be the kth-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.[14]

Secants and cosecants of sumsEdit

 

where   is the kth-degree elementary symmetric polynomial in the n 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.[15] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

 

Ptolemy's theoremEdit

 
Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine.

Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved (see the section on classical antiquity in the page History of trigonometry). It states that in a cyclic quadrilateral  , 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.[16] 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  .[17]

Multiple-angle formulaeEdit

Tn is the nth Chebyshev polynomial  [18]
de Moivre's formula, i is the imaginary unit  [19]

Multiple-angle formulaeEdit

Double-angle formulaeEdit

 
Visual demonstration of the double-angle formula for sine. The area, 1/2 × base × height, of an isosceles triangle is calculated, first when upright, and then on its side. When upright, the area =  . When on its side, the area = 1/2  . Rotating the triangle does not change its area, so these two expressions are equal. Therefore,  .

Formulae for twice an angle.[20]

 
 
 
 
 
 

Triple-angle formulaeEdit

Formulae for triple angles.[20]

 
 
 
 
 
 

Multiple-angle and half-angle formulaeEdit

 
 [21]

Chebyshev methodEdit

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the  th and  th values.[22]

  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 cos(x) 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 formulaeEdit

 

[23][24]

Also

 

TableEdit

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

Sine Cosine Tangent Cotangent
Double-angle formula[25][26]        
Triple-angle formula[18][27]        
Half-angle formula[23][24]        

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.[citation needed]

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x3 − 3x + d = 0, where   is the value of the cosine function at the one-third angle and d 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 formulaeEdit

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

Sine Cosine Other
     
     
     
     
 
Cosine power-reduction formula: an illustrative diagram. The red, orange and blue triangles are all similar, and the red and orange triangles are congruent. The hypotenuse   of the blue triangle has length  . The angle   is  , so the base   of that triangle has length  . That length is also equal to the summed lengths of   and  , i.e.  . Therefore,  , yielding the power-reduction formula when both sides are divided by  . The half-angle formula for cosine can be obtained by replacing   with   and taking the square-root of both sides.
 
Sine power-reduction formula: an illustrative diagram. The shaded blue and green triangles, and the red-outlined triangle   are all right-angled and similar, and all contain the angle  . The hypotenuse   of the red-outlined triangle has length  , so its side   has length  . The line segment   has length   and sum of the lengths of   and   equals the length of  , which is 1. Therefore,  . Subtracting   from both sides and dividing by 2 by two yields the power-reduction formula for sine. The half-angle formula for sine can be obtained by replacing   with   and taking the square-root of both sides. Note that this figure also illustrates, in the vertical line segment  , that  .

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[citation needed].

Cosine Sine
     
     

Product-to-sum and sum-to-product identitiesEdit

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.[28] 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[29]
 
 
 
 
 
 
 
 
Diagram illustrating sum-to-product identities for sine and cosine. The blue right-angled triangle has angle   and the red right-angled triangle has angle  . Both have a hypotenuse of length 1. Auxiliary angles, here called   and  , are constructed such that   and  . Therefore,   and  . This allows the two congruent purple-outline triangles   and   to be constructed, each with hypotenuse   and angle   at their base. The sum of the heights of the red and blue triangles is  , and this is equal to twice the height of one purple triangle, i.e.  . Writing   and   in that equation in terms of   and   yields the sum-to-product identity for sine. Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.
Sum-to-product[30]
 
 
 
 

Hermite's cotangent identityEdit

Charles Hermite demonstrated the following identity.[31] Suppose   are complex numbers, no two of which differ by an integer multiple of π. Let

 

(in particular,   being an empty product, is 1). Then

 

The simplest non-trivial example is the case n = 2:

 

Finite products of trigonometric functionsEdit

For coprime integers n, m

 

where Tn is the Chebyshev polynomial.

The following relationship holds for the sine function

 

More generally for an integer n > 0[32]

 

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 z on the complex unit circle and an integer n > 0,

 

Linear combinationsEdit

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 a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of   and  .

Sine and cosineEdit

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,[33][34]

 

where   and   are defined as so:

 

given that  

Arbitrary phase shiftEdit

More generally, for arbitrary phase shifts, we have

 

where   and   satisfy:

 

More than two sinusoidsEdit

The general case reads[34]

 

where

 

and

 

Lagrange's trigonometric identitiesEdit

These identities, named after Joseph Louis Lagrange, are:[35][36][37]

 
for  

A related function is the Dirichlet kernel:

 

Certain linear fractional transformationsEdit

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 functionEdit

Euler's formula states that, for any real number x:[38]

 
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,[39][40]

 
 

These formulae are useful for proving many other trigonometric identities. For example, that ei(θ+φ) = e e means that

cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).

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.

Function Inverse function[41]
   
   
   
   
   
   
   

Infinite product formulaeEdit

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[42][43]

 

Inverse trigonometric functionsEdit

The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[44]

 

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,[45]

 
 
 

Identities without variablesEdit

In terms of the arctangent function we have[45]

 

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 21/2 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:[46]

 
and so forth for all odd numbers, and hence
 

Many of those curious identities stem from more general facts like the following:[47]

 
and
 

Combining these gives us

 

If n 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 πEdit

An efficient way to compute π 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:[48][45]

 
 
 

Generally, for numbers t1, ..., tn−1 ∈ (−1, 1) for which θn = Σn−1
k=1
arctan tk ∈ (π/4, 3π/4)
, let tn = tan(π/2 − θn) = cot θn. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will be in (−1, 1). In particular, the computed tn will be rational whenever all the t1, ..., tn−1 values are rational. With these values,