List of trigonometric identities
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.
|Laws and theorems|
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.
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:
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. 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).
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.
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
This article uses the notation below for inverse trigonometric functions:
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.
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):
|in terms of|
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.
|versed sine, versine|
|versed cosine, vercosine|
|coversed sine, coversine|
|coversed cosine, covercosine|
|half versed sine, haversine|
|half versed cosine, havercosine|
|half coversed sine, hacoversine
|half coversed cosine, hacovercosine
|exterior secant, exsecant|
|exterior cosecant, excosecant|
Reflections, shifts, and periodicity
By examining the unit circle, the following properties of the trigonometric functions can be established.
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 formulas.
|θ reflected in α = 0
|θ reflected in α = π/||θ reflected in α = π/||θ reflected in α = π|
compare to α = 0
Shifts and periodicity
By shifting the arguments of trigonometric functions by certain angles, it is sometimes possible that changing the sign or applying complementary trigonometric functions can express particular results more simply. Some examples of shifts are shown below in the table.
- A full turn, or 360°, or 2π 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 180°, or π radian is the period of tan(x) = sin(x)/ and cot(x) = cos(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 they change the sign for a shift by any 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, 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 90°, or π/ 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/ 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/. 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 tan(x) and cot(x) by their quarter period (π/) does not yield such simple results.
|Shift by one quarter period||Shift by one half period||Shift by full periods||Period|
Angle sum and difference identities
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).
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. (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.
The sum and difference formulae for sine and cosine follow from the fact that a rotation of the plane by angle α, following a rotation by β, is equal to a rotation by α+β. 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 α with a column vector will rotate the column vector counterclockwise by the angle α.
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
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 sums
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.,
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.
and so on. The case of only finitely many terms can be proved by mathematical induction.
Secants and cosecants of sums
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. The case of only finitely many terms can be proved by mathematical induction on the number of such terms.
|Tn is the nth Chebyshev polynomial|
|de Moivre's formula, i is the imaginary unit|
Double-angle, triple-angle, and half-angle formulae
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 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
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
The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the (n − 1)th and (n − 2)th values.
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)x − x) = 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)x − x).
Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:
Tangent of an average
Setting either α or β to 0 gives the usual tangent half-angle formulae.
Viète's infinite product
(Refer to sinc function.)
Obtained by solving the second and third versions of the cosine double-angle formula.
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. 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.
Other related identities
- 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/ (right angle or quarter circle), then
Hermite's cotangent identity
(in particular, A1,1, being an empty product, is 1). Then
The simplest non-trivial example is the case n = 2:
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.)
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 cosine
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 shift
More generally, for an arbitrary phase shift, we have
More than two sinusoids
See also Phasor addition.
Lagrange's trigonometric identities
These identities, named after Joseph Louis Lagrange, are:
A related function is the following function of x, called the Dirichlet kernel.
Other sums of trigonometric functions
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 transformations
If f(x) is given by the linear fractional transformation
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.
Relation to the complex exponential function
With the unit imaginary number i satisfying i2 = −1,
These formulae are useful for proving many other trigonometric identities. For example, that ei(θ+φ) = eiθ eiφ 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 formulae
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:
The same cosine identity in radians is
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
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/ 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.
and so forth for all odd numbers, and hence
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:
or, alternatively, by using an identity of Leonhard Euler:
or by using Pythagorean triples:
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, q2 − p2, q2 + p2).
For example, for n = 3 terms,
for any a, b, c, d > 0.
A useful mnemonic for certain values of sines and cosines
For certain simple angles, the sines and cosines take the form √/ for 0 ≤ n ≤ 4, which makes them easy to remember.
With the golden ratio φ:
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:
Composition of trigonometric functions
where Ji are Bessel functions.
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:
verified using the identity tan x/ = 1 − cos x/. Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that (sin x)′ = cos x and (cos x)′ = −sin x. If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.
The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation:
The integral identities can be found in List of integrals of trigonometric functions. Some generic forms are listed below.
The fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and Fourier transforms.
Some differential equations satisfied by the sine function
Let i = √ be the imaginary unit and let ∘ denote composition of differential operators. Then for every odd positive integer n,
(When k = 0, then the number of differential operators being composed is 0, so the corresponding term in the sum above is just (sin x)n.) This identity was discovered as a by-product of research in medical imaging.
Further "conditional" identities for the case α + β + γ = 180°
The following formulae apply to arbitrary plane triangles and follow from α + β + γ = 180°, 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).
The "miraculous pentagram"
Let P, Q, R, S, T be the vertices of a pentagon on the surface of a unit sphere that are so situated that when their sides are extended to form a pentagram, they meet at right angles at the points of the star. Let
The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity:
The convolution of any integrable function of period 2π with the Dirichlet kernel coincides with the function's nth-degree Fourier approximation. The same holds for any measure or generalized function.
Tangent half-angle substitution
If we set
where eix = cos x + i sin x, sometimes abbreviated to cis x.
When this substitution of t for tan x/ is used in calculus, it follows that sin x is replaced by 2t/, cos x is replaced by 1 − t2/ and the differential dx is replaced by 2 dt/. Thereby one converts rational functions of sin x and cos x to rational functions of t in order to find their antiderivatives.
- Derivatives of trigonometric functions
- Exact trigonometric constants (values of sine and cosine expressed in surds)
- Half-side formula
- Hyperbolic function
- Laws for solution of triangles:
- List of integrals of trigonometric functions
- Proofs of trigonometric identities
- Pythagorean theorem
- Tangent half-angle formula
- Trigonometric constants expressed in real radicals
- Uses of trigonometry
- Versine and haversine
- Mnemonics in trigonometry
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- Values of sin and cos, expressed in surds, for integer multiples of 3° and of 5+5/°, and for the same angles csc and sec and tan