In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. 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 trick involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Sine redirects here. ... In mathematics, the term identity has several important uses: An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ... This article is about the concept of integrals in calculus. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

The unit circle

Notation

To avoid the confusion caused by the ambiguity of sin⁻¹(x), the reciprocals and inverses of trigonometric functions are often displayed as in this table. In representing the cosecant function, the longer form 'cosec' is sometimes used in place of 'csc'.

Different angular measures can be appropriate in different situations. This table shows some of the more common systems. Radians is the default angular measure and is the one you use if you use the exponential definitions. All angular measures are unitless. In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... The reciprocal function: y = 1/x. ...

This article describes the unit of angle. ... Some common angles, measured in radians. ... The grad is a measurement of plane angles of value 1/400 of a full circle, thus dividing a right angle in 100. ...

Basic relationship

Pythagorean trigonometric identity	[1]
Ratio identity	[1]

From the two identities above, the following table can be extrapolated. Note however that these conversion equations may not provide the correct sign (+ or −). For example, if sin θ = 1/2, the conversion in the table indicates that , though it is possible that . More information would be needed about which quadrant θ lies in to determine a single, exact answer. The Pythagorean trigonometric identity says that for any angle A: Proof Or: Note The reason for: is that any number, when divided by itself, is equal to one. ... It has been suggested that multiple sections of List of trigonometric identities be merged into this article or section. ...

Each trigonometric function in terms of the other five.

Function	sin	cos	tan	csc	sec	cot
sinθ =
cosθ =
tanθ =
cscθ =
secθ =
cotθ =

Historic shorthands

Rarely used today, the versine, coversine, haversine, and exsecant have been defined as below and used in navigation, for example the haversine formula was used to calculate the distance between two points on a sphere. The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the... In trigonometry, the coversine, denoted cvs(x), of an angle is defined as one minus the sine of the angle: The derivative of the coversine is the negative of the cosine and the integral is Very few applications of this function exist, and it is generally only used to provide... The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the... The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle. ... The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes. ...

Name	Value

Symmetry, shifts, and periodicity

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

Symmetry

When the trigonometric functions are reflected from certain values of θ, The result is often one of the other trigonometric functions. This leads to the following identities:

Reflected in θ = 0	Reflected in θ = π / 2	Reflected in θ = π

Shifts and periodicity

By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express the result more simply. Some examples of this are given shown by shifting functions round by π/2, π and 2π radians. Because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift.

Shift by π/2	Shift by π Period for tan and cot	Shift by 2π Period for sin, cos, csc and sec

Angle sum and difference identities

These are also known as the addition and subtraction theorems or formulæ. The quickest way to prove these is Euler's formula. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

Sine	[2]	Note: From Plus-Minus Sign. The plus-minus sign (±) is a mathematical symbol commonly used to indicate the precision of an approximation, or as a convenient shorthand for a quantity which has two possible values opposite in sign. ...
Cosine	[2]
Tangent	[2]

Sines and cosines of sums of infinitely many terms

In these two identities an asymmetry appears that is not seen in the case of sums of finitely many terms: in each product, there are only finitely many sine factors and cofinitely many cosine factors. In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. ...

If only finitely many of the terms θ_i are nonzero, then only finitely many of the terms on the right side will be nonzero because sine factors will vanish, and in each term, all but finitely many of the cosine factors will be unity.

Tangents of sums of finitely many terms

Let x_i = tan(θ_i ), for i = 1, ..., n. Let e_k be the kth-degree elementary symmetric polynomial in the variables x_i, i = 1, ..., n, k = 0, ..., n. Then In mathematics, specifically in commutative algebra, elementary symmetric polynomials are the basic building blocks for symmetric polynomials, in the sense that every symmetric polynomial can be expressed as a sum of products of the elementary symmetric polynomials. ...

the number of terms depending on n.

For example,

and so on. The general case can be proved by mathematical induction. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

Multiple-angle formulae

Tn is the nth Chebyshev polynomial	[3]
Sn is the nth spread polynomial
de Moivre's formula, i is the Imaginary unit

(This function of x is the Dirichlet kernel.) In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ... In the conventional language of trigonometry, the nth-degree spread polynomial Sn, for n = 0, 1, 2, ..., may be characterized by the trigonometric identity Although that is probably the simplest way to explain what spread polynomials are to those versed in well-known topics in mathematics, spread polynomials were introduced... de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematical analysis, the Dirichlet kernel is the collection of functions It is named after Johann Peter Gustav Lejeune Dirichlet. ...

Double-, triple-, and half-angle formulae

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

Double-angle formulae [4]
Triple-angle formulae [3]
Half-angle formulae [5]

See also Tangent half-angle formula. In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t. ...

Euler's infinite product

Power-reduction formulae

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

Sine
Cosine
Other

Product-to-sum and sum-to-product identities

The product-to-sum identies can be proven by expanding their right-hand sides using the angle addition theorems. In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

Product-to-sum

Sum-to-product

Other related identities

If x, y, and z are the three angles of any triangle, or in other words

(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.) In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ...

Ptolemy's theorem

(The first three equalities are trivial; the fourth is the substance of this identity.) Essentially this is Ptolemy's theorem adapted to the language of trigonometry. In mathematics, Ptolemys theorem is a relation in Euclidean geometry between the four sides and two diagonals or chords of a quadrilateral inscribed in circle. ...

Linear combinations

For some purposes it is important to know that any linear combination of sine waves of the same period but different phase shifts is also a sine wave with the same period, but a different phase shift. In the case of a linear combination of a sine and cosine wave, we have This article is about a portion of a periodic process. ...

where

More generally, for an arbitrary phase shift, we have

where

and

• note: arcsin, arccos, arctan are all inverses.

Other sums of trigonometric functions

Sum of sines and cosines with arguments in arithmetic progression:

For any a and b:

where arctan(y, x) is the generalization of arctan(y/x) which covers the entire circular range (see also the account of this same identity in "symmetry, periodicity, and shifts" above for this generalization of arctan).

The above identity is sometimes convenient to know when thinking about the Gudermanian function. Gudermannian function with its asymptotes y = ±π/2 marked in gray. ...

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

Inverse trigonometric functions

Compositions of trig and inverse trig functions

Relation to the complex exponential function

(Euler's formula),

where i² = −1. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

"cis"

It has been suggested that this section be split into a new article entitled cis (mathematics). (Discuss)

Occasionally one sees the notation Image File history File links No higher resolution available. ...

i.e. "cis" abbreviates "cos + i sin".

Though at first glance this notation is redundant, being equivalent to e^ix, its use is rooted in several advantages.

Convenience

This notation was more common in the post WWII era when typewriters were used to convey mathematical expressions. Superscripts are both offset vertically and smaller than 'cis' or 'exp'; hence, they can be problematic even for hand writing. For example e ^ix² versus cis( x²) versus exp( ix²). For many readers, cis( x²) is the clearest, easiest to read of the three.

The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas cis and cos + i sin notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for cos + i sin).

The cis notation is convenient for math students whose knowledge of trigonometry and complex numbers permit this notation, but whose conceptual understanding doesn't yet permit the notation e ^ix. As students learn concepts that build on prior knowledge, it is important not to force them into levels of math they are not yet prepared for.

Pedagogy

In some contexts, the cis notation may serve the pedagogical purpose of emphasizing that one has not yet proved that this is an exponential function. In doing trigonometry without complex numbers, one may prove the two identities

Similarly in treating multiplication of complex numbers (with no involvement of trigonometry), one may observe that the real and imaginary parts of the product of c₁ + is₁ and c₂ + is₂ are respectively

Thus one sees this same pattern arising in two disparate contexts:

• trigonometry without complex numbers, and
• complex numbers without trigonometry.

This coincidence can serve as a motivation for conjoining the two contexts and thereby discovering the trigonometric identity

and observing that this identity for cis of a sum is simpler than the identities for sin and cos of a sum. Having proved this identity, one can challenge the students to recall which familiar sort of function satisfies this same functional equation In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...

The answer is exponential functions. That suggests that cis may be an exponential function The exponential function is one of the most important functions in mathematics. ...

Then the question is: what is the base b? The definition of cis and the local behavior of sin and cos near zero suggest that

(where dx is an infinitesimal increment of x). Thus the rate of change at 0 is i, so the base should be eⁱ. Thus if this is an exponential function, then it must be Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

Infinite product formula

For applications to special functions, the following infinite product formulæ for trigonometric functions are useful: In mathematics, several functions are important enough to deserve their own name. ... In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ...

The Gudermannian function

The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers; see that article for details. The Gudermannian function, named after Christoph Gudermann (1798 - 1852), relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. ... Sine redirects here. ... A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions). ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Identities without variables

The curious identity Morries law is a name, that occasionally is used for the trigonometric identity It is a special case of the more general identity with and . ...

is a special case of an identity that contains one variable:

A similar-looking identity is

and in addition

The following is perhaps not as readily generalized to an identity containing variables:

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. In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ... This article is about the concept in number theory. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...

An efficient way to compute π is based on the following identity without variables, due to Machin: When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... John Machin, (1680—June 9, 1751), a professor of astronomy in London, is best known for developing a quickly converging series for Ï€ in 1706 and using it to compute Ï€ to 100 decimal places. ...

or, alternatively, by using Euler's formula: Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

With the golden ratio φ: For other uses, see Golden mean. ...

Also see exact trigonometric constants. Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. ...

Calculus

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, their derivatives can be found by verifying two limits. The first is: For other uses, see Calculus (disambiguation). ... Some common angles, measured in radians. ...

verified using the unit circle and squeeze theorem. It may be tempting to propose to use L'Hôpital's rule to establish this limit. However, if one uses this limit in order to prove that the derivative of the sine is the cosine, and then uses the fact that the derivative of the sine is the cosine in applying L'Hôpital's rule, one is reasoning circularly—a logical fallacy. The second limit is: Illustration of a unit circle. ... In calculus, the squeeze theorem (also known as the pinching theorem or the sandwich theorem, sometimes the squeeze lemma) is a theorem regarding the limit of a function. ... In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ...

verified using the identity tan(x/2) = (1 − cos(x))/sin(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. Series expansion redirects here. ...

The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation: This article is about derivatives and differentiation in mathematical calculus. ...

^[6]

The integral identities can be found in "list of integrals of trigonometric functions". The following is a list of integrals (antiderivative functions) of trigonometric functions. ...

Implications

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 transformations. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

Exponential definitions

Function	Inverse Function

Miscellaneous

Dirichlet kernel

The Dirichlet kernel D_n(x) is the function occurring on both sides of the next identity: In mathematical analysis, the Dirichlet kernel is the collection of functions It is named after Johann Peter Gustav Lejeune Dirichlet. ...

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. In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

Extension of half-angle formulae

If we set

then

and

and

where e^ix is the same as cis(x).

This substitution of t for tan(x/2), with the consequent replacement of sin(x) by 2t/(1 + t²) and cos(x) by (1 − t²)/(1 + t²) is useful in calculus for converting rational functions in sin(x) and cos(x) to functions of t in order to find their antiderivatives. For more information see tangent half-angle formula. For other uses, see Calculus (disambiguation). ... In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t. ...

See also

Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... It has been suggested that multiple sections of List of trigonometric identities be merged into this article or section. ... Trigonometry has an enormous variety of applications. ... In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t. ... Fig. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. ... The primary operation in differential calculus is finding a derivative. ...

References

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links