List of trigonometric identities

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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 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.

Notation[edit]

Angles[edit]

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 complete rotation (turn) = 360 degrees = 2π radians = 400 gons.

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
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.[2][3][4] 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 functions[edit]

Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin.

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.

Other functions[edit]

indicates the sign function, which is defined as:

Inverse functions[edit]

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 (arcsin or asin), satisfies

and

This article will denote the inverse of a trigonometric function by prefixing its name with "". The notation is shown in the table below.

The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians.

Original
function
Domain Image/range Inverse
function
Domain of
inverse
Range of usual
principal values of inverse

The symbol denotes the set of all real numbers. The set of all integer multiples of is denoted by

and the notation denotes the Minkowski sum of and the interval Explicitly,
and
where denotes set subtraction. In other words, the domain of and is the set of all real numbers that are not of the form for some integer Similarly, the domain of and is the set of all real numbers that do not belong to the set
said differently, it is the set of all real numbers that are not of the form for some integer

These inverse trigonometric functions are related to one another by the formulas:

which hold whenever they are well-defined (that is, whenever etc. are in the domains of the relevant functions).

Solutions to elementary trigonometric equations[edit]

The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values and all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some " is just another way of saying "for some integer "

Equation if and only if Solution where...
for some
for some
for some
for some
for some
for some

For example, if then for some While if then for some where is even if ; odd if The equations and have the same solutions as and respectively. In all equations above except for those just solved (i.e. except for / and /), for fixed and the integer in the solution's formula is uniquely determined by

The table below shows how two angles and must be related if their values under a given trigonometric function are equal or negatives of each other.

Equation if and only if Solution where... Also a solution to
for some
for some
for some
for some
for some
for some
for some

Pythagorean identities[edit]

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 (up to a plus or minus sign):

Each trigonometric function in terms of each of the other five.[5]
in terms of

Historical shorthands[edit]

All 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; however, this diagram is not exhaustive.

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[6][7]
(right) complementary angle, co-angle
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[edit]

Reflecting in ().

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

Reflections[edit]

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 x-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 x-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.[8]

reflected in [9]
odd/even identities
reflected in reflected in reflected in
compare to

Shifts and periodicity[edit]

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.

  • A full turn, or or radian leaves the unit circle fixed and is 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 or radian is the period of and as can be seen from these definitions and the period of the defining trigonometric functions. Therefore, shifting the arguments of and by any multiple of does not change their function values.
For the functions sin, cos, sec, and csc with period half a turn is half 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 so all together they change the sign for a shift by any odd multiple of that is, by with k an arbitrary integer. Any even multiple of is of course just a full period, and a backward shift by half a period is the same as a backward shift by one full period plus one shift forward by half a period.
  • A quarter turn, or or radian is a half-period shift for and with period (), yielding 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 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 The forward/backward shifts by one 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 and by their quarter period () does not yield such simple results.
Shift by one quarter period Shift by one half period[10] Shift by full periods[11] Period

Angle sum and difference identities[edit]

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

These are also known as the angle 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. The most intuitive derivation uses rotation matrices (see below).

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

For acute angles and , 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 ; 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.[12] (The diagram admits further variants to accommodate angles and sums greater than a right angle.) Dividing all elements of the diagram by provides yet another variant (shown) illustrating the angle sum formula for tangent.

These identities have applications in, for example, in-phase and quadrature components.

Illustration of the angle addition formula for the cotangent. Top right segment is of unit length.
Sine [13][14]
Cosine [14][15]
Tangent [14][16]
Cosecant [17]
Secant [17]
Cotangent [14][18]
Arcsine [19]
Arccosine [20]
Arctangent [21]
Arccotangent

Matrix form[edit]

The sum and difference formulae for sine and cosine follow from the fact that a rotation of the plane by angle following a rotation by is equal to a rotation by 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 (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 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[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.[22]

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

For example,

Multiple-angle formulae[edit]

Tn is the nth Chebyshev polynomial [24]
de Moivre's formula, i is the imaginary unit [25]

Double-angle, triple-angle, and half-angle formulae[edit]

Double-angle formulae[edit]

Formulae for twice an angle.[26]

Triple-angle formulae[edit]

Formulae for triple angles.[26]

Half-angle formulae[edit]

[27][28]

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 formulae[29][30]
Triple-angle formulae[24][31]
Half-angle formulae[27][28]

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 angles[edit]

For specific multiples, these follow from the angle addition formulae, while the general formula was given by 16th-century French mathematician François Viète.[citation needed]

for nonnegative values of k up through n.[citation needed]

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

[citation needed]

Chebyshev method[edit]

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the (n − 1)th and (n − 2)th values.[32]

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.

It follows by induction that cos(nx) is a polynomial of cos x, the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.

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 average[edit]

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

Viète's infinite product[edit]

(Refer to Viète's formula and sinc function.)

Power-reduction formulae[edit]

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 identities[edit]

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[33]
Sum-to-product[34]

Other related identities[edit]

  • [35]
  • 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 identity[edit]

Charles Hermite demonstrated the following identity.[36] 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: