List of trigonometric identities

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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[edit]

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 periodicity[edit]

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

Reflections[edit]

Unit circle with a swept angle theta plotted at coordinates (a,b). As the angle is reflected in increments of one-quarter pi (45 degrees), the coordinates are transformed. For a transformation of one-quarter pi (45 degrees, or 90 – theta), the coordinates are transformed to (b,a). Another increment of the angle of reflection by one-quarter pi (90 degrees total, or 180 – theta) transforms the coordinates to (-a,b). A third increment of the angle of reflection by another one-quarter pi (135 degrees total, or 270 – theta) transforms the coordinates to (-b,-a). A final increment of one-quarter pi (180 degrees total, or 360 – theta) transforms the coordinates to (a,-b).
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 periodicity[edit]

Unit circle with a swept angle theta plotted at coordinates (a,b). As the swept angle is incremented by one-half pi (90 degrees), the coordinates are transformed to (-b,a). Another increment of one-half pi (180 degrees total) transforms the coordinates to (-a,-b). A final increment of one-half pi (270 degrees total) transforms the coordinates to (b,a).
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

Signs[edit]

The sign of trigonometric functions depends on quadrant of the angle. If and sgn is the sign function,

The trigonometric functions are periodic with common period so for values of θ outside the interval they take repeating values (see § Shifts and periodicity above).

Angle sum and difference identities[edit]

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

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

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.

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 angles[edit]

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[edit]

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] The case of infinitely many terms can be proved by using some elementary inequalities.[15]

Secants and cosecants of sums[edit]

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.[16] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

Ptolemy's theorem[edit]

Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.

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.[17] 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 and half-angle formulae[edit]

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

Multiple-angle formulae[edit]

Double-angle formulae[edit]

Visual demonstration of the double-angle formula for sine. For the above isosceles triangle with unit sides and angle , the area 1/2 × base × height is calculated in two orientations. When upright, the area is . When on its side, the same area is . Therefore,

Formulae for twice an angle.[20]

Triple-angle formulae[edit]

Formulae for triple angles.[20]

Multiple-angle formulae[edit]

Formulae for multiple angles.[21]

Chebyshev method[edit]

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 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[edit]

[23][24]

Also

Table[edit]

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 are reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Power-reduction formulae[edit]

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, . Dividing both sides by yields the power-reduction formula for cosine: . 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.

if n is ...
n is odd
n is even

Product-to-sum and sum-to-product identities[edit]

Proof of the sum-and-difference-to-product cosine identity for prosthaphaeresis calculations using an isosceles triangle

The product-to-sum identities[28] 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.[29] 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[edit]

Sum-to-product identities[edit]

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 a sum-to-product identity for sine: . Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.

The sum-to-product identities are as follows:[30]

Hermite's cotangent identity[edit]

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 functions[edit]

For coprime integers n, m

where Tn is the Chebyshev polynomial.[citation needed]

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 combinations[edit]

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 cosine[edit]

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 shift[edit]

More generally, for arbitrary phase shifts, we have

where and satisfy:

More than two sinusoids[edit]

The general case reads[34]

where
and

Lagrange's trigonometric identities[edit]

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

for

A related function is the Dirichlet kernel:

A similar identity is[38]

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[edit]

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[edit]

Euler's formula states that, for any real number x:[39]

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,[40][41]

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[42]

Series expansion[edit]

When using a power series expansion to define trigonometric functions, the following identities are obtained:[43]

Infinite product formulae[edit]

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[44][45]

Inverse trigonometric functions[edit]

The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[46]

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,[47]

The arctangent function can be expanded as a series:[48]

Identities without variables[edit]

In terms of the arctangent function we have[47]

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:[49]

and so forth for all odd numbers, and hence

Many of those curious identities stem from more general facts like the following:[50]

and

Combining these gives us