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Trigonometric functions are differentiable. This is not immediately evident from the above geometrical definitions.

Moreover, the modern trend in mathematics is to build geometry from calculus rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus.

For defining trigonometric functions inside calculus, there are two equivalent possibilities, either using power series or differential equations.

These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.

Sine and cosine are the unique differentiable functions such that. Differentiating these equations, one gets that both sine and cosine are solutions of the differential equation.

Applying the quotient rule to the definition of the tangent as the quotient of the sine by the cosine, one gets that the tangent function verifies. Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions.

These recurrence relations are easy to solve, and give the series expansions [11]. The radius of convergence of these series is infinite.

Therefore, the sine and the cosine can be extended to entire functions also called "sine" and "cosine" , which are by definition complex-valued functions that are defined and holomorphic on the whole complex plane.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles.

Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions.

These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [14].

This identity can be proven with the Herglotz trick. For the proof of this expansion, see Sine.

From this, it can be deduced that. Euler's formula relates sine and cosine to the exponential function :. This formula is commonly considered for real values of x , but it remains true for all complex values.

Solving this linear system in sine and cosine, one can express them in terms of the exponential function:. When x is real, this may be rewritten as.

One can also define the trigonometric functions using various functional equations. For example, [16] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula.

By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.

Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities.

For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity.

One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.

The cosine and the secant are even functions ; the other trigonometric functions are odd functions. That is:.

This means that, for every integer k , one has. The Pythagorean identity , is the expression of the Pythagorean theorem in terms of trigonometric functions.

It is. The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves.

These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae. These identities can be used to derive the product-to-sum identities.

The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule.

The values given for the antiderivatives in the following table can be verified by differentiating them. The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function.

However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions.

To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function.

The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.

When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".

Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series.

They can also be expressed in terms of complex logarithms. See Inverse trigonometric functions for details. In this sections A , B , C denote the three interior angles of a triangle, and a , b , c denote the lengths of the respective opposite edges.

They are related by various formulas, which are named by the trigonometric functions they involve. The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C :.

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known.

This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

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