Trigonometric functions

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and are widely used for studying periodic phenomena through Fourier analysis.
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 commonly used. Each of these six trigonometric functions has a corresponding inverse function and has 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 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.

Notation

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 "" for sine, "" for cosine, "" or "" for tangent, "" for secant, "" or "" for cosecant, and "" or "" 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. 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 general functional notation in which
In contrast, the superscript 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.

Right-angled triangle definitions

If the acute angle is given, then any right triangles that have an angle of are similar to each other. This means that the ratio of any two side lengths depends only on. Thus these six ratios define six functions of, 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, and adjacent represents the side between the angle and the right angle.
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, or. 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.

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. One common unit is degrees, in which a right angle is 90° and a complete turn is 360°.
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 and can be defined for all complex numbers in terms of the exponential function, via power series, or as solutions to differential equations given particular initial values, without reference to any geometric notions. The other four trigonometric functions can be defined as quotients and reciprocals of and, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians 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, and a complete turn is an angle of 2 rad. Since radian is dimensionless, i.e. 1 rad = 1, the degree symbol can also be regarded as a mathematical constant factor such that 1° = /180 ≈ 0.0175.

Unit-circle definitions

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 of this coordinate system. While [|right-angled triangle definitions] allow for the definition of the trigonometric functions for angles between and radians 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 the positive half of the -axis. 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, is perpendicular to and intersects the - and -axes at points and The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of in the following manner.
The trigonometric functions and are defined, respectively, as the - and -coordinate values of point. That is,
and
In the range, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius 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.
File:Unit Circle Definitions of Six Trigonometric Functions.svg|thumb|upright=1.2|left|In this illustration, the six trigonometric functions of an arbitrary angle are represented as Cartesian coordinates of points related to the unit circle. The -axis ordinates of, and are, and, respectively, while the -axis abscissas of, and are, and, respectively.
The other trigonometric functions can also be found along the unit circle; all together, they are:
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, i.e.
File:Trigonometric functions.svg|right|thumb|upright=1.35|link=|Trigonometric functions:
–
Since a rotation of an angle of does not change the position or size of a shape, the points,,,, and 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 and any integer. 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. However, after a rotation by an angle, the points and already return to their original position, so that the tangent function and the cotangent function have a fundamental period of. That is, the equalities
and
hold for any angle and any integer.

Algebraic values

The algebraic expressions for the most important angles are as follows, beginning with the zero angle and ending with the right angle:
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.
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 a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of -th roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.
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.
If the sine of an angle is a rational number then the cosine is not necessarily a rational number, and vice-versa. However, if the tangent of an angle is rational then both the sine and cosine of the double angle will be rational.