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List of trigonometric identities

  (Redirected from Sum-to-product equation)
Cosines and sines around the unit circle

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.

Contents

NotationEdit

AnglesEdit

 
Signs of trigonometric functions in each quadrant. The mnemonic "All Science Teachers (are) Crazy" lists the basic functions ('All', sin, tan, cos) which are positive from quadrants I to IV.[1] This is a variation on the mnemonic "All Students Take Calculus".

This article uses Greek letters such as alpha (α), beta (β), gamma (γ), and theta (θ) to represent angles. Several different units of angle measure are widely used, including degree, radian, and gradian (gons):

1 full circle (turn) = 360 degree = 2π radian = 400 gon.

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:

Conversions of common angles
Turn Degree Radian Gradian sine cosine tangent
             
             
             
             
             
             
             
             
             
             
             
             
             
             
             
             
             

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.[2][3] The analogous condition for the unit radian requires that the argument divided by π is rational, and yields the solutions 0, π/6, π/2, 5π/6, π, 7π/6, 3π/2, 11π/6(, 2π).

Trigonometric functionsEdit

The functions sine, cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. Their usual abbreviations are sin(θ), cos(θ) and tan(θ), respectively, where θ denotes the angle. The parentheses around the argument of the functions are often omitted, e.g., sin θ and cos θ, 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 hypotenuse).

 

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 sin and cos from above:

 

The remaining trigonometric functions secant (sec), cosecant (csc), and cotangent (cot) 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 functionsEdit

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 (sin−1) or arcsine (arcsin or asin), satisfies

 

and

 

This article uses the notation below for inverse trigonometric functions:

Function sin cos tan sec csc cot
Inverse arcsin arccos arctan arcsec arccsc arccot

Pythagorean identitiesEdit

In trigonometry, the basic relationship between the sine and the cosine is given by the Pythagorean identity:

 

where sin2 θ means (sin(θ))2 and cos2 θ means (cos(θ))2.

This can be viewed as a version of the Pythagorean theorem, and follows from the equation x2 + y2 = 1 for the unit circle. This equation can be solved for either the sine or the cosine:

 

where the sign depends on the quadrant of θ.

Dividing this identity by either sin2 θ or cos2 θ 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 (up to a plus or minus sign):

Each trigonometric function in terms of the other five.[4]
in terms of            
             
             
             
             
             
             

Historical shorthandsEdit

 
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Many of these terms are no longer in common use.

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.

Name Abbreviation Value[5][6]
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 periodicityEdit

 
Reflecting θ in α=0 (α=π)

By examining the unit circle, the following properties of the trigonometric functions can be established.

ReflectionsEdit

When a direction, represented by an angle   enclosed with the x-direction, is reflected in a line with direction   then the angle   of this reflected direction has the value

 

This way, reflections in the directions 0 and π radian ( and 180°) generate equally looking results (see picture). 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.

θ reflected in α = 0[7]
odd/even identities
θ reflected in α = π/4
co-function identities[8]
θ reflected in α = π/2 θ reflected in α = π
compare to α = 0
       
       
       
       
       
       

Shifts and periodicityEdit

By shifting round the arguments of trigonometric functions by certain angles, it is sometimes possible that changing the sign or applying complementary trigonometric functions express particular results more simply. Some examples of shifts are shown below in the table.

  • A full turn, or 360°, or 2π radian does not change anything along the unit circle and makes up the smallest interval for which the trigonometric functions sin, cos, sec, and csc repeat their values, and is thus their period. Shifting arguments of any periodic function by any integer multiple of a full period preserves the function value of the unshifted argument.
  • A half turn, or 180°, or π radian is the period of tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x), as can be seen from these definitions and the period of the defining trigonometric functions. So shifting the arguments of tan(x) and cot(x) by any multiple of π, does not change their function values.
For the functions sin, cos, sec, and csc with period 2π half a turn is half of their period. For this shift they change the sign of their values, as can be seen from the unit circle again. This new value repeats after any additional shift of 2π, so all together for a shift by an odd multiple of π, i.e., by (2k + 1)⋅π, with k an arbitrary integer. Any even multiple of π is of course just a full period. A backward shift by half a period is represented by a backward shift by one full period and one shift forward by half a period.
  • A quarter turn, or 90°, or π/2 radian is a half period shift for tan(x) and cot(x) with period π (180°), and yields the function value of applying the complementary function to the unshifted argument. By the argument above this also holds for a shift by any odd multiple (2k + 1)⋅π/2 of the half period.
For the four other trigonometric functions a quarter turn also represents a quarter period. A shift by an arbitrary multiple of a quarter period, that is not covered by a multiple of half periods, can be decomposed in an integer multiple of periods, plus or minus one quarter period. The terms expressing these multiples are (4k + 1)⋅π/2 and (4k - 1)⋅π/2. These forward/backward shifts by a quarter period are reflected in the table below. Again, these shifts yield function values, employing the respective complementary function applied to the unshifted argument.
Shifting the arguments of tan(x) and cot(x) by their quarter period (π/4) does not yield such simple results.
Shift by one quarter period Shift by one half period[9] Shift by full periods[10] Period
       
       
       
       
       
       

Angle sum and difference identitiesEdit

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

These are also known as the addition and subtraction theorems or formulae. 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. Furthermore, it is even possible to derive the identities using Euler's identity although this would be a more obscure approach given that complex numbers are used.

 
Illustration of the angle addition formula for the tangent. Emphasized segments are of unit length.

For acute angles α and β, whose sum is non-obtuse, a concise diagram (shown) 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 β; the opposite and adjacent legs for this angle have respective lengths sin β and cos β. The cos β leg is itself the hypotenuse of a right triangle with angle α; that triangle's legs, therefore, have lengths given by sin α and cos α, multiplied by cos β. The sin β leg, as hypotenuse of another right triangle with angle α, likewise leads to segments of length cos α sin β and sin α sin β. Now, we observe that the "1" segment is also the hypotenuse of a right triangle with angle α + β; the leg opposite this angle necessarily has length sin(α + β), while the leg adjacent has length cos(α + β). 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.[11] (The diagram admits further variants to accommodate angles and sums greater than a right angle.) Dividing all elements of the diagram by cos α cos β provides yet another variant (shown) illustrating the angle sum formula for tangent.

 
Illustration of the angle addition formula for the cotangent. Top right segment is of unit length.
Sine  [12][13]
Cosine  [13][14]
Tangent  [13][15]
Cotangent  [13][16]
Arcsine  [17]
Arccosine  [18]
Arctangent  [19]
atan2  
Arccotangent  

Matrix formEdit

The sum and difference formulae for sine and cosine can be written in matrix form as:

 

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 (technically, the special orthogonal group SO(2)), since the composition law is fulfilled and inverses exist. Furthermore, matrix multiplication of the rotation matrix for an angle α with a column vector will rotate the column vector counterclockwise by the angle α.

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 θi 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 ek (for k = 0, 1, 2, 3, ...) be the kth-degree elementary symmetric polynomial in the variables

 

for i = 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.[20]

Secants and cosecants of sumsEdit

 

where ek is the kth-degree elementary symmetric polynomial in the n variables xi = tan θi, i = 1, ..., 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.[21] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

 

Multiple-angle formulaeEdit

Tn is the nth Chebyshev polynomial    [22]
Sn is the nth spread polynomial  
de Moivre's formula, i is the imaginary unit      [23]

Double-angle, triple-angle, and half-angle formulaeEdit

Double-angle formulaeEdit

 
 
 
 
 
 

Triple-angle formulaeEdit

 
 
 
 
 
 

Half-angle formulaeEdit

 
 
 
 
 
 

[24][25]

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 formulae[26][27]        
Triple-angle formulae[22][28]        
Half-angle formulae[24][25]        

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 4x3 − 3x + d = 0, where x 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.

Sine, cosine, and tangent of multiple anglesEdit

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 k up through n.

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 methodEdit

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the (n − 1)th and (n − 2)th values.[29]

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)xx) = cos((n − 1)x) cos x + sin((n − 1)x) sin x.

Similarly, sin(nx) can be computed from sin((n − 1)x), sin((n − 2)x), and cos(x) 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)xx).

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

 

Tangent of an averageEdit

 

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

Viète's infinite productEdit

 

(Refer to sinc function.)

Power-reduction formulaeEdit

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

Sine Cosine Other
     
     
     
     

and in general terms of powers of sin θ or cos θ 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. 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[30]
 
 
 
 
 
 
Sum-to-product[31]
 
 
 

Other related identitiesEdit

  •  [32]
  • If x + y + z = π (half circle), then
     
  • Triple tangent identity: If x + y + z = π (half circle), then
     
In particular, the formula holds when x, y, and z are the three angles of any triangle.
(If any of x, y, z is a right angle, one should take both sides to be . This is neither +∞ nor −∞; for present purposes it makes sense to add just one point at infinity to the real line, that is approached by tan θ as tan θ either increases through positive values or decreases through negative values. This is a one-point compactification of the real line.)
  • Triple cotangent identity: If x + y + z = π/2 (right angle or quarter circle), then
     

Hermite's cotangent identityEdit

Charles Hermite demonstrated the following identity.[33] Suppose a1, ..., an are complex numbers, no two of which differ by an integer multiple of π. Let

 

(in particular, A1,1, being an empty product, is 1). Then

 

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

 

Ptolemy's theoremEdit

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

If w + x + y + z = π, then:
 

(The first three equalities are trivial rearrangements; the fourth is the substance of this identity.)

Finite products of trigonometric functionsEdit

For coprime integers n, m

 

where Tn is the Chebyshev polynomial.

The following relationship holds for the sine function

 

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 c and φ.

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,[34][35][36]

 

where the original amplitudes a and b sum in quadrature to yield the combined amplitude c,

 

and, using the atan2 function, the initial value of the phase angle x + φ is obtained by

 

Arbitrary phase shiftEdit

More generally, for an arbitrary phase shift, we have

 

where

 

and

 

More than two sinusoidsEdit

The general case reads[37]

 

where

 

and

 

See also Phasor addition.

Lagrange's trigonometric identitiesEdit

These identities, named after Joseph Louis Lagrange, are:[38][39]

 

A related function is the following function of x, called the Dirichlet kernel.

 

see proof.

Other sums of trigonometric functionsEdit

Sum of sines and cosines with arguments in arithmetic progression:[40] if α ≠ 0, then

 

For any a and b:

 

where atan2(y, x) is the generalization of arctan(y/x) that covers the entire circular range.

 

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 x, y, and z are the three angles of any triangle, i.e. if x + y + z = π, then

 

Certain linear fractional transformationsEdit

If f(x) is given by the linear fractional transformation

 

and similarly

 

then

 

More tersely stated, if for all α we let fα be what we called f above, then

 

If x is the slope of a line, then f(x) is the slope of its rotation through an angle of α.

Inverse trigonometric functionsEdit

 
 [41]

Compositions of trig and inverse trig functionsEdit

 

Relation to the complex exponential functionEdit

With the unit imaginary number i satisfying i2 = −1,

 [42] (Euler's formula),
 
  (Euler's identity),
 
 [43]
 [44]
 

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.

Infinite product formulaeEdit

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[45][46][citation needed]

 

Identities without variablesEdit

In terms of the arctangent function we have[41]

 

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 x = 15,

 
 

For the case x = 10,

 

The same cosine identity is

 

Similary,

 
 
 

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:[47]

 
 
 

and so forth for all odd numbers, and hence

 

Many of those curious identities stem from more general facts like the following:[48]

 

and

 

Combining these gives us

 

If n is an odd number (n = 2m + 1) 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 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

 [49][41]
 [49]
 [41]

Generally, for numbers t1, ..., tn−1 ∈ (−1, 1) for which θn = ∑n−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,

 

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 tk values is not within (−1, 1). Note that when t = p/q is rational then the (2t, 1 − t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2p2, q2 + p2).

For example for n = 3 terms,

 

for any a, b, c, d > 0.

A useful mnemonic for certain values of sines and cosinesEdit

For certain simple angles, the sines and cosines take the form n/2 for 0 ≤ n ≤ 4, which makes them easy to remember.

 

MiscellanyEdit

With the golden ratio φ:

 
 

Also see trigonometric constants expressed in real radicals.

An identity of EuclidEdit

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 functionsEdit

This identity involves a trigonometric function of a trigonometric function:[50]

 
 
 
 

where Ji are Bessel functions.

CalculusEdit

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: