The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example sin(x). Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression would typically be interpreted to mean so parentheses are required to express
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example and denote not This differs from the (historically later) general functional notation in which
However, the exponent is commonly used to denote the inverse function, not the reciprocal. For example and denote the inverse trigonometric function alternatively written The equation implies not In this case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than are not in common use.
In this right triangle, denoting the measure of angle BAC as A: sin A = a/c; cos A = b/c; tan A = a/b.Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin.
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ, and adjacent represents the side between the angle θ and the right angle.[3][4]
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π/2radians. Therefore and represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
Top: Trigonometric function sin θ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants. Bottom: Graph of sine function versus angle. Angles from the top panel are identified.
Summary of relationships between trigonometric functions[5]
In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics).
However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function, via power series,[6] or as solutions to differential equations given particular initial values[7] (see below), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions ifthe argument is regarded as an angle given in radians.[6] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions.[8] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2π (≈ 6.28) rad. For real number x, the notations sin x, cos x, etc. refer to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/π)°, so that, for example, sin π = sin 180° when we take x = π. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π/180 ≈ 0.0175.
All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O.Sine function on unit circle (top) and its graph (bottom)In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle. The ordinates of A, B and D are sin θ, tan θ and csc θ, respectively, while the abscissas of A, C and E are cos θ, cot θ and sec θ, respectively.Signs of trigonometric functions in each quadrant. Mnemonics like "allstudents take calculus" indicates when sine, cosine, and tangent are positive from quadrants I to IV.[9]
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and radians(90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let be the ray obtained by rotating by an angle θ the positive half of the x-axis (counterclockwise rotation for and clockwise rotation for ). This ray intersects the unit circle at the point The ray extended to a line if necessary, intersects the line of equation at point and the line of equation at point The tangent line to the unit circle at the point A, is perpendicular to and intersects the y- and x-axes at points and The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A. That is,
In the range , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse. And since the equation holds for all points on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.
The other trigonometric functions can be found along the unit circle as
and
and
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
Since a rotation of an angle of does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of . Thus trigonometric functions are periodic functions with period . That is, the equalities
and
hold for any angle θ and any integerk. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that is the smallest value for which they are periodic (i.e., is the fundamental period of these functions). However, after a rotation by an angle , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of . That is, the equalities
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.[11]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
For an angle which, measured in degrees, is a multiple of three, the exact trigonometric values of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by ruler and compass.
For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.
For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.
Graphs of sine, cosine and tangentThe sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.Animation for the approximation of cosine via Taylor polynomials. together with the first Taylor polynomials
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number.[12] Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Hardy describes four ways of defining the trigonometric functions:
Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically.
By a power series, which is particularly well-suited to complex variables.
By using an infinite product expansion.
By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.
To this list of ways of defining the trigonometric functions, Bartle and Sherbert add:
Sine is the unique solution with y(0) = 0 and y′(0) = 1; cosine is the unique solution with y(0) = 1 and y′(0) = 0.
One can then prove, as a theorem, that solutions are periodic, having the same period. Writing this period as is then a definition of the real number which is independent of geometry.
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
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. Here, the poles are the numbers of the form for the tangent and the secant, or for the cotangent and the cosecant, where k is an arbitrary integer.
The following infinite product for the sine is due to Leonhard Euler, and is of great importance in complex analysis:[20]
This may be obtained from the partial fraction decomposition of given above, which is the logarithmic derivative of .[21] From this, it can be deduced also that
This formula is commonly considered for real values of x, but it remains true for all complex values.
Proof: Let and One has for j = 1, 2. The quotient rule implies thus that . Therefore, is a constant function, which equals 1, as This proves the formula.
One has
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
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups.[22] The set of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group , via an isomorphism In pedestrian terms , and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number (the base), the function defines an isomorphism of the group . The real and imaginary parts of are the cosine and sine, where is used as the base for measuring angles. For example, when , we get the measure in radians, and the usual trigonometric functions. When , we get the sine and cosine of angles measured in degrees.
Note that is the unique value at which the derivative becomes a unit vector with positive imaginary part at . This fact can, in turn, be used to define the constant .
The unit circle is a rational curve under the substitution[23] which covers every point once except . Then the element of arc length along the circle is Consider the integral The integrand has poles at imaginary points , with residues . It follows that the integral is independent of the complex contour of integration only if taken to have values in the Riemann surface. The factor of two allows us to define as the semiperiod: The trigonometric functions are defined by inverting . Namely, we put for a point of the graph of the function . This argument also shows that the cosine and sine, defined in this way, have semiperiod .
Note that the substitution transforms the integrand defining to itself while exchanging the intervals and , and so the quarter period is Using this, one can then show that by making the rational substitution in the integral defining . Hence we have Likewise,
More complicated substitutions of this kind can also be used to show the addition formulae
By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
The cosine and sine functions are periodic, with period , which is the smallest positive period: Consequently, the secant and cosecant also have as their period. The functions sine and cosine also have semiperiods , and It therefore follows that as well as other identities such as We also have The function has a unique zero (at ) in the strip . The function has the pair of zeros in the same domain. Because of the periodicity, the zeros of sine are There zeros of cosine are All of the zeros are simple zeros, and each function has derivative at each of the zeros.
The tangent function has a simple zero at and vertical asymptotes at , where it has a simple pole of residue . Again, owing to the periodicity, the zeros are all the integer multiples of and the poles are odd multiples of , all having the same residue. The poles correspond to vertical asymptotes
The cotangent function has a simple pole of residue 1 at the integer multiples of and simple zeros at odd multiples of . The poles correspond to vertical asymptotes
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π/2], see Proofs 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.
All trigonometric functions are periodic functions of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one has
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.
Sum
Difference
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration.