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Fundamental IdentitiesIf an equation contains one or more variables and is valid for all replacement values of thevariables for which both sides of the equation are defined, then the equation is known as anidentity. The equationx2 + 2x =x(x + 2), for example, is an identity because it is valid for all

replacement values ofx.

If an equation is valid only for certain replacement values of the variable, then it is called aconditional equation. The equation 3x + 4 = 25, for example, is a conditional equation becauseit is not valid for all replacement values ofx. An equation that is said to be an identity withoutstating any restrictions is, in reality, an identity only for those replacement values for which bothsides of the identity are defined. For example, the identity

is valid only for those values of for which both sides of the equation are defined.

The fundamental (basic) trigonometric identities can be divided into several groups. First are thereciprocal identities. These include

Next are the quotient identities. These include
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Then there are the cofunction identities. These include

Next there are the identities for negatives. These include

Finally there are the Pythagorean identities. These include

The second identity is obtained by dividing the first by cos2 , and the third identity is obtainedby dividing the first by sin2 . The process of showing the validity of one identity based onpreviously known facts is called proving the identity. The validity of the foregoing identitiesfollows directly from the definitions of the basic trigonometric functions and can be used toverify other identities.

No standard method for solving identities exists, but there are some general rules or strategiesthat can be followed to help guide the process:

1. Try to simplify the more complicated side of the identity until it is identical to the secondside of the identity.

2. Try to transform both sides of an identity into an identical third expression.

3. Try to express both sides of the identity in terms of only sines and cosines; then try tomake both sides identical.

4. Try to apply the Pythagorean identities as much as possible.

5. Try to use factoring and combining of terms, multiplying one side of the identity by anexpression that is equal to 1, squaring both sides of the identity, and other algebraictechniques to manipulate equations.


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Example 1: Use the basic trigonometric identities to determine the other five values of thetrigonometric functions given that

Example 2: Verify the identity cos + sin tan = sec .


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Example 3: Verify the identity

Example 4: Verify the identity


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11 Other sums of trigonometric functions
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12 Certain linear fractional transformations

13 Inverse trigonometric functions

13.1 Compositions of trig and inverse trig functions

14 Relation to the complex exponential function

15 Infinite product formulae

16 Identities without variables

16.1 Computing

16.2 A useful mnemonic for certain values of sines and cosines

16.3 Miscellany

16.4 An identity of Euclid

17 Calculus

17.1 Implications

18 Exponential definitions 19 Miscellaneous

19.1 Dirichlet kernel

19.2 Weierstrass substitution

20 See also

21 Notes

22 References

23 External links

[edit] Notation[edit] Angles

This article uses Greek letters such asalpha (),beta (), gamma(), and theta () to representangles. Several different units of angle measureare widely used, including degrees,radians, andgrads:

1 full circle = 360 degrees = 2 radians = 400 grads.

The following table shows the conversions for some common angles:

Degre

es30 60 120 150 210 240 300 330

Radia

ns

Grads33

grad

66

grad

133

grad

166

grad

233

grad

266

grad

333

grad

366

grad
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Degre

es45 90 135 180 225 270 315 360

Radians

Grads50

grad

100

grad

150

grad

200

grad

250

grad

300

grad

350

grad

400

grad

Unless otherwise specified, all angles in this article are assumed to be in radians, though anglesending in a degree symbol () are in degrees.

[edit] Trigonometric functions

The primary trigonometric functions are the sineand cosineof an angle. These are sometimesabbreviated sin() and cos(), respectively, where is the angle, but the parentheses around theangle are often omitted, e.g., sin and cos .

The tangent (tan) of an angle is theratio of the sine to the cosine:

Finally, the reciprocal functionssecant (sec), cosecant (csc), and cotangent (cot) are thereciprocals of the cosine, sine, and tangent:

These definitions are sometimes referred to as ratio identities.

[edit] Inverse functions

Main article: Inverse trigonometric functions

The inverse trigonometric functions are partialinverse functions for the trigonometric functions.For example, the inverse function for the sine, known as the inverse sine (sin1) orarcsine(arcsin or asin), satisfies

and

This article uses the following notation for inverse trigonometric functions:
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Functi

onsin cos tan sec csc cot

Invers

e

arcsi

n

arcc

os

arcta

n

arcs

ec

arcc

sc

arcco

t

[edit] Pythagorean identityThe basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:

where cos2means (cos())2 and sin2means (sin())2.

This can be viewed as a version of the Pythagorean theorem, and follows from the equationx2 +y2 = 1 for the unit circle. This equation can be solved for either the sine or the cosine:

[edit] Related identities

Dividing the Pythagorean identity through by either cos2or sin2yields two other identities:

Using these identities together with the ratio identities, it is possible to express any trigonometricfunction in terms of any other (up toa plus or minus sign):

Each trigonometric function in terms of the other five.[1]

in

terms

of
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[edit] Historic shorthands

All of the trigonometric functions of an angle can be constructed geometrically in

terms of a unit circle centered at O. Many of these terms are no longer in common

use.

The versine,coversine,haversine, and exsecant were used in navigation. For example thehaversine formula was used to calculate the distance between two points on a sphere. They arerarely used today.

Name(s) Abbreviation(s) Value[2]

versed sine, versine 1 cos()
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versed cosine, vercosine 1 + cos()

coversed sine, coversine 1 sin()

coversed cosine, covercosine 1 + sin()

half versed sine, haversine

half versed cosine,

havercosine

half coversed sine,

hacoversine

cohaversine

half coversed cosine,

hacovercosine

cohavercosine

exterior secant, exsecant sec() 1

exterior cosecant,

excosecantcsc() 1

chord

[edit] Symmetry, shifts, and periodicityBy examining the unit circle, the following properties of the trigonometric functions can be

established.

[edit] Symmetry

When the trigonometric functions are reflected from certain angles, the result is often one of theother trigonometric functions. This leads to the following identities:

Reflected in =

0[3]Reflected in = / 2

(co-function

Reflected in =
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identities)[4]

[edit] Shifts and periodicity

By shifting the function round by certain angles, it is often possible to find differenttrigonometric functions that express the result more simply. Some examples of this are shown by

shifting functions round by /2, and 2 radians. Because the periods of these functions areeither or 2, there are cases where the new function is exactly the same as the old functionwithout the shift.

Shift by /2

Shift by

Period for tan and

cot[5]

Shift by 2

Period for sin, cos, csc

and sec[6]

[edit] Angle sum and difference identitiesThese are also known as the addition and subtraction theorems orformul. They were originallyestablished by the 10th century Persian mathematician Ab al-Waf' Bzjn. One method of

proving these identities is to apply Euler's formula. The use of the symbols and is describedin the articleplus-minus sign.

Sine[7][8]

Cosine[8][9]
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Tangent[8][10]

Arcsine[11]

Arccosin

e [12]

Arctang

ent[13]

[edit] Matrix form

See also: matrix multiplication

The sum and difference formulae for sine and cosine can be written inmatrix form as:

This shows that these matrices form a representationof the rotation group in the plane(technically, the special orthogonal groupSO(2)), since the composition law is fulfilled:subsequent multiplications of a vector with these two matrices yields the same result as therotation by the sum of the angles.

[edit] Sines and cosines of sums of infinitely many terms
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In these two identities an asymmetry appears that is not seen in the case of sums of finitely manyterms: in each product, there are only finitely many sine factors andcofinitely many cosinefactors.

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

[edit] Tangents of sums of finitely many terms

Let ek (fork {0, ..., n}) be the kth-degreeelementary symmetric polynomial in the variables

fori {0, ..., n}, i.e.,

Then

the number of terms depending on n.

For example:
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and so on. The general case can be proved by mathematical induction.[14]

[edit] Secants and cosecants of sums of finitely many terms

where ek is the kth-degree elementary symmetric polynomial in the n variablesxi = tan i,i = 1, ..., n, and the number of terms in the denominator depends on n.

For example,

[edit] Multiple-angle formulae

Tn is the nth Chebyshev

polynomial [15]

Sn is the nth spread polynomial
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de Moivre's formula, iis the

imaginary unit [16

]

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

See also:Tangent half-angle formula

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

Double-angle formulae[17][18]

Triple-angle formulae[15][19]

Half-angle formulae[20][21]
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The fact that the triple-angle formula for sine and cosine only involves powers of a singlefunction allows one to relate the geometric problem of acompass and straightedge constructionofangle trisection to the algebraic problem of solving a cubic equation, which allows one toprove that this is in general impossible, by field theory.

A formula for computing the trigonometric identities for the third-angle exists, but it requires

finding the zeroes of the cubic equation , wherex is the value of the sine

function at some angle and dis the known value of the sine function at the triple angle. However,the discriminant of this equation is negative, so this equation has three real roots (of which onlyone is the solution within the correct third-circle) but none of these solutions is reducible to a realalgebraic expression, as they use intermediate complex numbers under thecube roots, (whichmay be expressed in terms of real-only functions only if using hyperbolic functions).
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[edit] Sine, cosine, and tangent of multiple angles

For specific multiples, these follow from the angle addition formulas, while the general formulawas given by 16th century French mathematicianVieta.

In each of these two equations, the first parenthesized term is abinomial coefficient, and the finaltrigonometric function equals one or minus one or zero so that half the entries in each of thesums are removed. Tan ncan be written in terms of tan using the recurrence relation:

cot ncan be written in terms of cot using the recurrence relation:

[edit] Chebyshev method

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula

knowing the (n 1)th and (n 2)th formulae.[22]

The cosine fornx can be computed from the cosine of (n 1)x and (n 2)x as follows:

Similarly sin(nx) can be computed from the sines of (n 1)x and (n 2)x

For the tangent, we have:

whereH/K= tan(n 1)x.
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[edit] Tangent of an average

Setting either orto 0 gives the usual tangent half-angle formul.

[edit] Vite's infinite product

[edit] Power-reduction formulaObtained by solving the second and third versions of the cosine double-angle formula.

Sine Cosine Other

and in general terms of powers of sinor costhe following is true, and can be deduced usingDe Moivre's formula, Euler's formula andbinomial theorem.

Cosine Sine

if

nis

od

d
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if

n

is

ev

en

[edit] Product-to-sum and sum-to-product identitiesThe product-to-sum identities orprosthaphaeresis formulascan be proven by expanding theirright-hand sides using the angle addition theorems. Seebeat (acoustics)for an application of thesum-to-product formul.

Product-to-sum[23] Sum-to-product[24]

[edit] Other related identities

Ifx,y, andzare the three angles of any triangle, or in other words

(If any ofx,y,zis 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 thereal line, that isapproached by tan() as tan() either increases through positive values or decreases throughnegative values. This is a one-point compactification of the real line.)
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[edit] Hermite's cotangent identity

Main article: Hermite's cotangent identity

Charles Hermite demonstrated the following identity.[25]Suppose a1, ..., an arecomplex numbers,no two of which differ by an integer multiple of. Let

(in particular,A1,1, being anempty product, is 1). Then

The simplest non-trivial example is the case n = 2:

cot(z a1)cot(z a2) = 1 + cot(a1 a2)cot(z a1) + cot(a2 a1)cot(z

a2).

[edit] Ptolemy's theorem

(The first three equalities are trivial; the fourth is the substance of this identity.) Essentially thisis Ptolemy's theorem adapted to the language of modern trigonometry.

[edit] Linear combinationsFor some purposes it is important to know that any linear combination of sine waves of the sameperiod or frequency but differentphase shifts is also a sine wave with the same period orfrequency, but a different phase shift. In the case of a non-zero linear combination of a sine andcosine wave[26] (which is just a sine wave with a phase shift of /2), we have

where
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=25http://en.wikipedia.org/wiki/Hermite's_cotangent_identityhttp://en.wikipedia.org/wiki/Charles_Hermitehttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=26http://en.wikipedia.org/wiki/Ptolemy's_theoremhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=27http://en.wikipedia.org/wiki/Phase_(waves)http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=25http://en.wikipedia.org/wiki/Hermite's_cotangent_identityhttp://en.wikipedia.org/wiki/Charles_Hermitehttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=26http://en.wikipedia.org/wiki/Ptolemy's_theoremhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=27http://en.wikipedia.org/wiki/Phase_(waves)


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or equivalently

or even

More generally, for an arbitrary phase shift, we have

where

and

For the most general case, see Phasor addition.

[edit] Lagrange's trigonometric identitiesThese identities, named afterJoseph Louis Lagrange, are:[27][28]

A related function is the following function ofx, called the Dirichlet kernel.
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=28http://en.wikipedia.org/wiki/Joseph_Louis_Lagrangehttp://en.wikipedia.org/wiki/Joseph_Louis_Lagrangehttp://en.wikipedia.org/wiki/Dirichlet_kernelhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=28http://en.wikipedia.org/wiki/Joseph_Louis_Lagrangehttp://en.wikipedia.org/wiki/Dirichlet_kernel


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[edit] Other sums of trigonometric functionsSum of sines and cosines with arguments in arithmetic progression[29]:

For any a and b:

where atan2(y,x) is the generalization ofarctan(y/x) that covers the entire circular range.

The above identity is sometimes convenient to know when thinking about the Gudermannianfunction, which relates the circularand hyperbolic trigonometric functions without resorting tocomplex numbers.

Ifx,y, andzare the three angles of any triangle, i.e. ifx +y +z= , then

[edit] Certain linear fractional transformationsIf(x) is given by the linear fractional transformation
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=29http://en.wikipedia.org/wiki/Atan2http://en.wikipedia.org/wiki/Gudermannian_functionhttp://en.wikipedia.org/wiki/Gudermannian_functionhttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Hyperbolic_functionhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=30http://en.wikipedia.org/wiki/M%C3%B6bius_transformationhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=29http://en.wikipedia.org/wiki/Atan2http://en.wikipedia.org/wiki/Gudermannian_functionhttp://en.wikipedia.org/wiki/Gudermannian_functionhttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Hyperbolic_functionhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=30http://en.wikipedia.org/wiki/M%C3%B6bius_transformation


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and similarly

then

More tersely stated, if for all we let be what we called above, then

Ifx is the slope of a line, then (x) is the slope of its rotation through an angle of .

[edit] Inverse trigonometric functions

[edit] Compositions of trig and inverse trig functions

[edit] Relation to the complex exponential function
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=31http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=32http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=33http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=31http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=32http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=33


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[30] (Euler's formula),

ei = 1 (Euler's identity),

[31]

[32]

and hence the corollary:

where i2 = 1.

[edit] Infinite product formulaeFor applications to special functions, the following infinite product formulae for trigonometricfunctions are useful:[33][34]

[edit] Identities without variablesThe curious identity
http://en.wikipedia.org/wiki/Euler's_formulahttp://en.wikipedia.org/wiki/Euler's_identityhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=34http://en.wikipedia.org/wiki/Special_functionshttp://en.wikipedia.org/wiki/Infinite_producthttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=35http://en.wikipedia.org/wiki/Morrie's_lawhttp://en.wikipedia.org/wiki/Euler's_formulahttp://en.wikipedia.org/wiki/Euler's_identityhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=34http://en.wikipedia.org/wiki/Special_functionshttp://en.wikipedia.org/wiki/Infinite_producthttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=35http://en.wikipedia.org/wiki/Morrie's_law


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

Similarly:

A similar-looking identity is

Similarly:

The following is perhaps not as readily generalized to an identity containing variables (but seeexplanation 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 than21/2 that are relatively primeto (or have noprime factors in common with) 21. The last several

examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosinesare the real parts of the zeroes of those polynomials; the sum of the zeroes is theMbius functionevaluated at (in the very last case above) 21; only half of the zeroes are present above. The twoidentities preceding this last one arise in the same fashion with 21 replaced by 10 and 15,respectively.

A lot of those curious identities stem from more general facts like the following [35]:
http://en.wikipedia.org/wiki/Coprimehttp://en.wikipedia.org/wiki/Coprimehttp://en.wikipedia.org/wiki/Prime_factorhttp://en.wikipedia.org/wiki/Cyclotomic_polynomialhttp://en.wikipedia.org/wiki/M%C3%B6bius_functionhttp://en.wikipedia.org/wiki/M%C3%B6bius_functionhttp://en.wikipedia.org/wiki/Coprimehttp://en.wikipedia.org/wiki/Prime_factorhttp://en.wikipedia.org/wiki/Cyclotomic_polynomialhttp://en.wikipedia.org/wiki/M%C3%B6bius_function


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and

Combining these gives us

Ifn is an odd number (n = 2m + 1) we can make use of the symmetries to get

[edit] Computing

An efficient way to compute is based on the following identity without variables, due toMachin:

or, alternatively, by using an identity ofLeonhard Euler:

[edit] 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 makesthem easy to remember.
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=36http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/John_Machinhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=37http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=36http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/John_Machinhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=37


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[edit] Miscellany

With the golden ratio:

Also see exact trigonometric constants.

[edit] An identity of Euclid

Euclid showed in Book XIII, Proposition 10 of hisElements that the area of the square on theside of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares onthe sides of the regular hexagon and the regular decagon inscribed in the same circle. In thelanguage of modern trigonometry, this says:

Ptolemy used this proposition to compute some angles in his table of chords.

[edit] CalculusIn calculusthe relations stated below require angles to be measured in radians; the relationswould become more complicated if angles were measured in another unit such as degrees. If thetrigonometric functions are defined in terms of geometry, their derivatives can be found byverifying two limits. The first is:
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verified using the unit circleand squeeze theorem. The second limit is:

verified using the identity tan(x/2) = (1 cosx)/sinx. Having established these two limits, onecan use the limit definition of the derivative and the addition theorems to show that (sinx) = cosx and (cosx) = sinx. If the sine and cosine functions are defined by theirTaylor 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 therules ofdifferentiation:[36][37][38]

The integral identities can be found in "list of integrals of trigonometric functions". Some genericforms are listed below.
http://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Squeeze_theoremhttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functionshttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Squeeze_theoremhttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions


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[edit] Implications

The fact that the differentiation of trigonometric functions (sine and cosine) results in linearcombinations of the same two functions is of fundamental importance to many fields ofmathematics, including differential equationsandFourier transforms.

[edit] Exponential definitionsFunction Inverse function[39]
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[edit] Miscellaneous

[edit] Dirichlet kernel

The Dirichlet kernelDn(x) is the function occurring on both sides of the next identity:

The convolution of any integrable functionof period 2 with the Dirichlet kernel coincides withth

Weierstrass substitution

Main article: Weierstrass substitution

If we set

then[40]

where eix = cos(x) + i sin(x), sometimes abbreviated to cis(x).

When this substitution oftfor tan(x/2) is used in calculus, it follows that sin(x) is replaced by 2t/(1 + t2), cos(x) is replaced by (1 t2)/(1 + t2) and the differential dx is replaced by (2 dt)/(1 + t2).Thereby one converts rational functions of sin(x) and cos(x) to rational functions oftin order tofind their antiderivatives.
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=43http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=44http://en.wikipedia.org/wiki/Dirichlet_kernelhttp://en.wikipedia.org/wiki/Convolutionhttp://en.wikipedia.org/wiki/Integrable_functionhttp://en.wikipedia.org/wiki/Integrable_functionhttp://en.wikipedia.org/wiki/Weierstrass_substitutionhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=43http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=44http://en.wikipedia.org/wiki/Dirichlet_kernelhttp://en.wikipedia.org/wiki/Convolutionhttp://en.wikipedia.org/wiki/Integrable_functionhttp://en.wikipedia.org/wiki/Weierstrass_substitutionhttp://en.wikipedia.org/wiki/Calculus