NationMaster - Encyclopedia: Trigonometric identity

FACTOID # 158: 84% of people in Finland feel that they are at a low risk of experiencing a burglary - but just look at how many burglaries they have!

Interesting crime facts »

Home

Encyclopedia

WHAT'S NEW

RECENT ARTICLES

More Recent Articles »

SEARCH ALL

FACTS & STATISTICS Advanced view

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Trigonometric identity

In mathematics, trigonometric identities are equations involving 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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

Notation

The following notations hold for all six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). For brevity, only the sine case is given in the table. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ... Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ... Secant is a term in mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Notation	Reading	Description	Definition
sin²(x)	"sine squared [of] x"	the square of sine; sine to the second power	sin²(x) = (sin(x))²
arcsin(x)	"arcsine [of] x"	the inverse function for sine	arcsin(x) = y if and only if sin(y) = x and $-{pi over 2} le y le {pi over 2}$
(sin(x))⁻¹	"sine [of] x, to the negative-one power"	the reciprocal of sine; the multiplicative inverse of sine	(sin(x))⁻¹ = 1 / sin(x)

arcsin(x) can also be written sin⁻¹(x); this must not be confused with (sin(x))⁻¹. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... â†” â‡” â‰¡ logical symbols representing iff. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...

Definitions

$tan (x) = frac {sin (x)} {cos(x)} qquad operatorname{cot}(x) = frac {cos (x)} {sin(x)} = frac{1} {tan(x)}$

$operatorname{sec}(x) = frac{1} {cos(x)} qquad operatorname{csc}(x) = frac{1} {sin(x)}$

For more information, including definitions based on the sides of a right triangle, see Trigonometric functions. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Periodicity, symmetry, and shifts

These are most easily shown from the unit circle:

Periodicity (for any integer k)	Symmetry	Shifts
$sin(x) = sin(x + 2kpi) ,$	$sin(-x) = -sin(x) ,$	$sin(x) = cosleft(frac{pi}{2} - xright)$
$cos(x) = cos(x + 2kpi) ,$	$cos(-x) =; cos(x) ,$	$cos(x) = sinleft(frac{pi}{2}-xright)$
$tan(x) = tan(x + kpi) ,$	$tan(-x) = -tan(x) ,$	$tan(x) = cotleft(frac{pi}{2} - xright)$
	$cot(-x) = -cot(x) ,$

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 other words, we have Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ...

$asin x+bcos x=sqrt{a^2+b^2}cdotsin(x+varphi)$

where

$varphi= left{ begin{matrix} {rm arctan}(b/a),&&mbox{if }age0; ; pi-{rm arctan}(b/a),&&mbox{if }a<0. ; end{matrix} right. ;$

Pythagorean identities

These identities are based on the Pythagorean theorem. The first is sometimes simply called the Pythagorean trigonometric identity. The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ... 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. ...

$sin^2(x) + cos^2(x) = 1 ;$

$tan^2(x) + 1 = sec^2(x) ;$

$1 + cot^2(x) = csc^2(x) ;$

Note that the second equation is obtained from the first by dividing both sides by cos²(x). To get the third equation, divide the first by sin²(x) instead.

Angle sum and difference identities

These are also known as the addition and subtraction theorems or formulae. The quickest way to prove these is Euler's formula. The tangent formula follows from the other two. A geometric proof of the sin(x + y) identity is given at the end of this article. Error creating thumbnail: convert: unable to open image `/mnt/upload3/wikipedia/commons/e/eb/Eulers_formula. ...

$sin(x pm y) = sin(x) cos(y) pm cos(x) sin(y),$

$cos(x pm y) = cos(x) cos(y) mp sin(x) sin(y),$

$tan(x pm y) = frac{tan(x) pm tan(y)}{1 mp tan(x)tan(y)}$

${rm cdot{imath} s}(x+y)={rm cdot{imath} s}(x),{rm cdot{imath} s}(y)$

${rm cdot{imath} s}(x-y)={{rm cdot{imath} s}(x)over{rm cdot{imath} s}(y)}$

where

${rm cdot{imath} s}(x)=exp(i x)=e^{i x}=cos(x)+i sin(x),$

and

$i =sqrt{-1}.,$

See also Ptolemaios' theorem. In mathematics, Ptolemaios theorem is a relation in Euclidean geometry between the four sides and two diagonals or chords of a quadrilateral inscribed in circle. ...

Double-angle formulae

These can be shown by substituting x = y in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivre's formula with n = 2. De Moivres formula states that for any real number x and any integer n, The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

$sin(2x) = 2 sin (x) cos(x) ,$

$cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x) ,$

$tan(2x) = frac{2 tan (x)} {1 - tan^2(x)}$

The double-angle formulae can also be used to find Pythagorean triples. If (a, b, c) are the lengths of the sides of a right triangle, then (a² − b², 2ab, c²) also form a right triangle, where angle B is the angle being doubled. If a² − b² is negative, take its opposite and use the supplement of B. The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A pair of angles are supplementary if their respective measures sum to 180 degrees. ...

Triple-angle formulae

$cos(3x)= 4 cos^3(x) - 3 cos(x) ,$

$sin(3x)= 3 sin(x)- 4 sin^3(x) ,$

$tan(3x)= frac{3 tan x - tan^3 x}{1 - 3 tan^2(x)}$

Multiple-angle formulae

If T_n is the nth Chebyshev polynomial then In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (ÐŸÐ°Ñ„Ð½ÑƒÑ‚Ð¸Ð¹ Ð§ÐµÐ±Ñ‹ÑˆÑ‘Ð²), are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ...

$cos(nx)=T_n(cos(x)). ,$

de Moivre's formula: De Moivres formula states that for any real number x and any integer n, The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

$cos(nx)+isin(nx)=(cos(x)+isin(x))^n ,$

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. ...

$1+2cos(x)+2cos(2x)+2cos(3x)+cdots+2cos(nx) = frac{ sinleft(left(n+frac{1}{2}right)xright) }{ sin(x/2) }.$

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. For the computer science usage see convolution (computer science) . 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... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

Power-reduction formulae

Solve the second and third versions of the cosine double-angle formula for cos²(x) and sin²(x), respectively.

$cos^2(x) = {1 + cos(2x) over 2}$

$sin^2(x) = {1 - cos(2x) over 2}$

$sin^2(x) cos^2(x) = {1 - cos(4 x) over 8}$

$sin^3(x) = frac{3 sin(x) - sin(3 x)}{4}$

$cos^3(x) = frac{3 cos(x) + cos(3 x)}{4}$

Half-angle formulae

Sometimes the formulae in the previous section are called half-angle formulae. To see why, substitute x/2 for x in the power reduction formulae, then solve for cos(x/2) and sin(x/2) to get:

$cosleft(frac{x}{2}right) = pm, sqrt{frac{1 + cos(x)}{2}}$

$sinleft(frac{x}{2}right) = pm, sqrt{frac{1 - cos(x)}{2}}$

These may also be called the half-angle formulae. Then

$tanleft(frac{x}{2}right) = {sin (x/2) over cos (x/2)} = pm, sqrt{1 - cos x over 1 + cos x}. qquad qquad (1)$

Multiply both numerator and denominator inside the radical by 1 + cos x, then simplify (using a Pythagorean identity):

$tanleft(frac{x}{2}right) = pm, sqrt{(1 - cos x) (1 + cos x) over (1 + cos x) (1 + cos x)} = pm, sqrt{1 - cos^2 x over (1 + cos x)^2}$

$= {sin x over 1 + cos x}.$

Likewise, multiplying both numerator and denominator inside the radical — in equation (1) — by
1 − cos x, then simplifying:

$tanleft(frac{x}{2}right) = pm, sqrt{(1 - cos x) (1 - cos x) over (1 + cos x) (1 - cos x)} = pm, sqrt{(1 - cos x)^2 over (1 - cos^2 x)}$

$= {1 - cos x over sin x}.$

Thus, the pair of half-angle formulae for the tangent are:

$tanleft(frac{x}{2}right) = frac{sin(x)}{1 + cos(x)} = frac{1-cos(x)}{sin(x)}.$

If we set

$t = tanleft(frac{x}{2}right),$

then

$sin(x) = frac{2t}{1 + t^2}$

and

$cos(x) = frac{1 - t^2}{1 + t^2}$

and

$e^{i x} = frac{1 + i t}{1 - i t}.$

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. Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... 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. ...

Product-to-sum identities

These 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. ...

$cosleft (xright ) cosleft (yright ) = {cosleft (x + yright ) + cosleft (x - yright ) over 2} ;$

$sinleft (xright ) sinleft (yright ) = {cosleft (x - yright ) - cosleft (x + yright ) over 2} ;$

$sinleft (xright ) cosleft (yright ) = {sinleft (x - yright ) + sinleft (x + yright ) over 2} ;$

Sum-to-product identities

Replace x by (x + y) / 2 and y by (x – y) / 2 in the product-to-sum formulae.

$cos(x) + cos(y) = 2 cosleft( frac{x + y}{2} right) cosleft( frac{x - y}{2} right) ;$

$sin(x) + sin(y) = 2 sinleft( frac{x + y}{2} right) cosleft( frac{x - y}{2} right) ;$

$cos(x) - cos(y) = -2 sinleft( {x + y over 2}right) sinleft({x - y over 2}right) ;$

$sin(x) - sin(y) = 2 cosleft({x + yover 2}right) sinleft({x - yover 2}right) ;$

Additionally we have for any a and b:

$a cos(x) + b sin(x) = sqrt{ a^2 + b^2 } cos(x - arctan(b, a)) ;$

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).

Inverse trigonometric functions

$arcsin(x)+arccos(x)=pi/2;$

$arctan(x)+arccot(x)=pi/2.;$

$arctan(x)+arctan(1/x)=left{begin{matrix} pi/2, & mbox{if }x > 0 -pi/2, & mbox{if }x < 0 end{matrix}right..$

$arctan(x)+arctan(y)=arctanleft(frac{x+y}{1-xy}right) ;$

$arctan(x)-arctan(y)=arctanleft(frac{x-y}{1+xy}right) ;$

$sin(arccos(x))=sqrt{1-x^2} ,$

$cos(arcsin(x))=sqrt{1-x^2} ,$

$sin(arctan(x))=frac{x}{sqrt{1+x^2}}$

$cos(arctan(x))=frac{1}{sqrt{1+x^2}}$

$tan(arcsin (x))=frac{x}{sqrt{1 - x^2}}$

$tan(arccos (x))=frac{sqrt{1 - x^2}}{x}$

Every trigonometric function can be related directly to every other trigonometric function. Such relations can be expressed by means of inverse trigonometric functions as follows: let φ and ψ represent a pair of trigonometric functions, and let arcψ be the inverse of ψ, such that ψ(arcψ(x))=x. Then φ(arcψ(x)) can be expressed as an algebraic formula in terms of x. Such formulae are shown in the table below: φ can be made equal to the head of one of the rows, and ψ can be equated to the head of a column:

*Table of conversion formulae*
φ / ψ	sin	cos	tan	csc	sec	cot
sin		$sqrt{1 - x^2}$	${x over sqrt{1 + x^2}}$		${sqrt{x^2 - 1} over x}$	${1 over sqrt{1 + x^2}}$
cos	$sqrt{1 - x^2}$		${1 over sqrt{1 + x^2}}$	${sqrt{x^2 - 1} over x}$		${x over sqrt{1 + x^2}}$
tan
csc		${1 over sqrt{1 - x^2}}$	${sqrt{1 + x^2} over x}$		${x over sqrt{x^2 - 1}}$	$sqrt{1 + x^2}$
sec	${1 over sqrt{1 - x^2}}$		$sqrt{1 + x^2}$	${x over sqrt{x^2 - 1}}$		${sqrt{1 + x^2} over x}$
cot

One procedure that can be used to obtain the elements of this table is as follows:
Given trigonometric functions φ and ψ, what is φ(arcψ(x)) equal to?

Find an equation that relates φ(u) and ψ(u) to each other:
Let u = arc ψ(x), so that:
Solve the last equation for φ(arcψ(x)).

Example. What is cot(arccsc(x)) equal to? First, find an equation which relations the functions cot and csc to each other, such as

Second, let u = arccsc(x):

Third, solve this equation for cot(arccsc(x)):

and this is the formula which shows up in the sixth row and fourth column of the table.

Exponential forms

where

Infinite product formulae

For applications to special functions, the following infinite product formulae 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. Gudermannian function with its asymptotes y = ±π/2 marked in gray. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Identities without variables

Richard Feynman is reputed to have learned as a boy, and always remembered, the following curious identity: Richard Phillips Feynman (May 11, 1918 â€“ February 15, 1988) (surname pronounced FINE-man; in IPA) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics. ...

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

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. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... 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 MÃ¶bius 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: Lower-case pi The mathematical constant Ï€ is a real number which is defined as the ratio of a circles circumference (Greek Ï€ÎµÏÎ¹Ï†ÎÏÎµÎ¹Î±, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... John Machin, (1680â€”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. ...

Under this heading, there are also "special values" of trigonometric functions, including the ones that every student of trigonometry learns:

With the golden ratio φ: The golden ratio, also known as the golden proportion, golden mean, golden section, golden number, divine proportion or sectio divina, is an irrational number, approximately 1. ...

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, then their derivatives can be found by verifying two limits. The first is: Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... The radian (symbol: rad) is the SI unit of plane angle. ...

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 sandwich theorem) is a theorem regarding the limit of a function. ... In calculus, lHÃ´pitals rule (alternately, lHospitals 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. As the degree of the Taylor series rises, it approaches the correct function. ...

The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation. We have: In mathematics, the derivative is one of the two central concepts of calculus. ...

The integral identities can be found in Wikipedia's table of integrals. Integration is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. ...

Geometric proofs

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

In the figure the angle x is part of right angled triangle ABC, and the angle y part of right angled triangle ACD. Then construct DG perpendicular to AB and construct CE parallel to AB. Image File history File links Sinesum. ...

Angle x = Angle BAC = Angle ACE = Angle CDE.

EG = BC.

cos(x + y) = cos(x) cos(y) − sin(x) sin(y)

Using the above figure:

External links

A one page proof of many trigonometric identities using Euler's formula, by Connelly Barnes.

Categories: Mathematical identities | Trigonometry Error creating thumbnail: convert: unable to open image `/mnt/upload3/wikipedia/commons/e/eb/Eulers_formula. ...

Results from FactBites:

Trigonometric identity - Wikipedia, the free encyclopedia (1318 words)
	In mathematics, trigonometric identities are equations involving 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.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Notation GA_googleFillSlot("encyclopedia_square");

Definitions

Periodicity, symmetry, and shifts

Pythagorean identities

Angle sum and difference identities

Double-angle formulae

Triple-angle formulae

Multiple-angle formulae

Power-reduction formulae

Half-angle formulae

Product-to-sum identities

Sum-to-product identities

Inverse trigonometric functions

Exponential forms

Infinite product formulae

The Gudermannian function

Identities without variables

Calculus

Geometric proofs

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

cos(x + y) = cos(x) cos(y) − sin(x) sin(y)

See also

External links

Notation