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The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. For extending these definitions to functions whose domain is the whole projectively extended real line, one can use geometrical definitions using the standard unit circle (a circle with radius 1 unit). Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of the sine and the cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.
Right-angled triangle definitions
Plot of the six trigonometric functions and the unit circle for an angle of 0.7 radians
In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lower case letter denotes an edge of the triangle and its length.
Given an acute angleA = ? of a right-angled triangle, the hypotenuseh is the side that connects the two acute angles. The side badjacent to ? is the side of the triangle that connects ? to the right angle. The third side a is said opposite to ?.
If the angle ? is given, then all sides of the right-angled triangle are well defined up to a scaling factor. This means that the ratio of any two side lengths depends only on ?. These six ratios define thus six functions of ?, which are the trigonometric functions. More precisely, the six trigonometric functions are:
In a right angled triangle, the sum of the two acute angles is a right angle, that is 90° or radians.
Summary of relationships between trigonometric functions
Top: Trigonometric function sin ? for selected angles ?, ? - ?, ? + ?, and 2? - ? in the four quadrants. Bottom: Graph of sine function versus angle. Angles from the top panel are identified.
Radians versus degrees
In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in degrees.
When using trigonometric function in calculus, their argument is generally not an angle, but rather a real number. In this case, it is more suitable to express the argument of the trigonometric as the length of the arc of the unit circle delimited by an angle with the center of the circle as vertex. Therefore, one uses the radian as angular unit: a radian is the angle that delimits an arc of length on the unit circle. A complete turn is thus an angle of 2? radians.
A great advantage of radians is that many formulas are much simpler when using them, typically all formulas relative to derivatives and integrals.
This is thus a general convention that, when the angular unit is not explicitly specified, the arguments of trigonometric functions are always expressed in radians.
In this illustration, the six trigonometric functions of an arbitrary angle ? are represented as Cartesian coordinates of points related to the unit circle. The ordinates of A, B and D are sin ?, tan ? and csc ?, respectively, while the abscissas of A, C and E are cos ?, cot ? and sec ?, respectively.
Signs of trigonometric functions in each quadrant. The mnemonic "allscience teachers (are) crazy" lists the functions which are positive from quadrants I to IV. This is a variation on the mnemonic "All Students Take Calculus".
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 permit the definition of the trigonometric functions for angles between 0 and radian(90°), the unit circle definitions allow to extend the domain of the trigonometric functions to all positive and negative real numbers.
Rotating a ray from the direction of the positive half of the x-axis by an angle ? (counterclockwise for and clockwise for ) yields intersection points of this ray (see the figure) with the unit and, by extending the ray to a line if necessary, with the and with the The tangent line to the unit circle in point A, which is orthogonal to this ray, intersects the y- and x-axis in points and . The coordinate values of these points give all the existing values of the trigonometric functions for arbitrary real values of ? in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A, i.e.,
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 for all points on the unit circle the equation holds, this definition of cosine and sine also satisfies the Pythagorean identity
The other trigonometric functions can be found along the unit circle as
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
As 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
hold for any angle ? and any integerk. The same is true for the four other trigonometric functions. Observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, shows that 2? is the smallest value for which they are periodic, i.e., 2? is the fundamental period of these functions. However, already after a rotation by an angle the points B and C return to their original position, so that the tangent function and the cotangent function have a fundamental period of ?. That is, the equalities
hold for any angle ? and any integer k.
The unit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.
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 proving that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.
For an angle which, measured 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.
Simple algebraic values
The following table summarizes the simplest algebraic values of trigonometric functions. The symbol ∞ represents the point at infinity on the projectively extended real line; it is not signed, because, when it appears in the table, the corresponding trigonometric function tends to +∞ on one side, and to -∞ on the other side, when the argument tends to the value in the table.
The 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
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.
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
Power series expansion
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
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.
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 , as This proves the formula.
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.
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.
Trigonometric functions in the complex plane
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, ?/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 2?. This is the smallest period, except for the tangent and the cotangent, which have ? 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.
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
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.
The notations sin-1, cos-1, etc. are often used for arcsin and arccos, etc. 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".
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.
Law of sines
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:
where ? is the area of the triangle,
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.
In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem.
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (k = 1) have additional nodes. The oscillation seen about the sawtooth when k is large is called the Gibbs phenomenon
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.
Under rather general conditions, a periodic function f(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series. Denoting the sine or cosine basis functions by ?k, the expansion of the periodic function f(t) takes the form:
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).
The word sine derives from Latinsinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khw?rizm? into Medieval Latin.
The choice was based on a misreading of the Arabic written form j-y-b (), which itself originated as a transliteration from Sanskrit j?v?, which along with its synonym jy? (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek "string".
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans--"cutting"--since the line cuts the circle.
The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.
All Students Take Calculus -- a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane