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Wikipedia / Sine and cosine

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse.

The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the sajyā and sakoṭi-jyā functions used in Indian astronomy during the Gupta period.

Elementary descriptions

Right-angled triangle definition

, start with a right triangle is the angle of interest. The three sides of the triangle are named as follows:

The opposite side.
The hypotenuse. The hypotenuse is always the longest side of a right-angled triangle.
The adjacent side. It forms a side of (and is adjacent to) both the angle of interest and the right angle.

Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:

The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The reciprocal of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as:

Special angle measures

. However, this is not the case as all such triangles are similar, and so the ratios are the same for each of them. For example, each leg.. The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator.

Laws

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. is the triangle's circumradius.

The law of cosines is useful for computing the length of an unknown side if two other sides and an angle are known., the resulting equation becomes the Pythagorean theorem.

Vector definition

The cross product and dot product are operations on two vectors in Euclidean vector space, then sine and cosine can be defined as:

Analytic descriptions

Unit circle definition

The sine and cosine functions may also be defined in a more general way by using unit circle in the Cartesian coordinate system with the positive half of the axis intersects the unit circle at exactly one point. The and , respectively; that is,

because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the , even under the new definition using the unit circle.

Graph of a function and its elementary properties

, the point is rotated counterclockwise and stopped exactly on the , the point returns to its origin. This results in both sine and cosine functions having the range.

Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the coordinate. In other words, both sine and cosine functions are periodic, meaning any angle added by the circle's circumference is the angle itself. Mathematically,

is said to be odd, and is said to be even. The sine function is odd, whereas the cosine function is even. Both sine and cosine functions are similar, with their difference being shifted

This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts:

Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function. The only real fixed point of the cosine function is called the Dottie number. The decimal expansion of the Dottie number is approximately 0.739085.

Continuity and differentiation

The sine and cosine functions are infinitely differentiable. The derivative of sine is cosine, and the derivative of cosine is negative sine: Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself. These derivatives can be applied to the first derivative test, according to which the monotonicity of a function can be defined as the inequality of function's first derivative greater or less than equal to zero. It can also be applied to second derivative test, according to which the concavity of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero. The following table shows that both sine and cosine functions have concavity and monotonicity) is decreasing (going downward)in certain intervals. This information can be represented as a Cartesian coordinates system divided into four quadrants. to the two-dimensional system of differential equations with the initial conditions. One could interpret the unit circle in the above definitions as defining the phase space trajectory.

Integral and the usage in mensuration

Their area under a curve can be obtained by using the integral denotes the constant of integration. These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the arc length is the incomplete elliptic integral of the second kind. It cannot be expressed using elementary functions. is the gamma function is the lemniscate constant.

Inverse functions

would have an arcsine of zero). To define the inverse functions of sine and cosine, they must be restricted to their principal branches. for arccosine.

Other identities

According to Pythagorean theorem, the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the Pythagorean trigonometric identity, the sum of a squared sine and a squared cosine equals 1:

Sine and cosine satisfy the following double-angle formulas:

The cosine double angle formula implies that and are, themselves, shifted and scaled sine waves. Specifically, The graph shows both sine and sine squared functions, with the sine in blue and the sine squared in red. Both graphs have the same shape but with different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods.

Series and polynomials

Both sine and cosine functions can be defined by using a Taylor series, a power series involving the higher-order derivatives. As mentioned in , the derivative, continuing to repeat those four functions. The . One can then use the theory of Taylor series to show that the following identities hold for all real numbers is the angle in radians. More generally, for all complex numbers: Taking the derivative of each term gives the Taylor series for cosine:

Both sine and cosine functions with multiple angles may appear as their linear combination, resulting in a polynomial. Such a polynomial is known as the trigonometric polynomial. The trigonometric polynomial's ample applications may be acquired in its interpolation, and its extension of a periodic function known as the Fourier seriesis defined as:

The trigonometric series be any coefficients, then the trigonometric series can be defined as:, the coefficients of a trigonometric series are:

Complex numbers relationship

Complex exponential function definitions

Both sine and cosine can be extended further via complex number, a set of numbers composed of both real and imaginary numbers, the definition of both sine and cosine functions can be extended in a complex plane in terms of an exponential function as follows:

Alternatively, both functions can be defined in terms of Euler's formula:

When plotted on the complex plane traces out the unit circle as:

, both sine and cosine functions can be expressed in terms of real sines, cosines, and hyperbolic functions as:

Polar coordinates

Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates.

Complex arguments

are the hyperbolic sine and cosine. These are entire functions.

Partial fraction and product expansions of complex sine

Using the partial fraction expansion technique in complex analysis

Usage of complex sine

is found in the functional equation for the Gamma function which in turn is found in the functional equation for the Riemann zeta-function

As a holomorphic function, is a 2D solution of Laplace's equation

The complex sine function is also related to the level curves of pendulums.

Complex graphs

Background

Etymology

The word sine is derived, indirectly, from the Sanskrit word 'bow-string' or more specifically its synonym (both adopted from Ancient Greek 'string; chord'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see jyā, koti-jyā and utkrama-jyā; sine and chord are closely related in a circle of unit diameter, see Ptolemy's Theorem). This was transliterated in Arabic as , which is meaningless in that language and written as (). Since Arabic is written without short vowels, was interpreted as the homograph (arجيبno), which means 'bosom', 'pocket', or 'fold'. When the Arabic texts of Al-Battani and al-Khwārizmī were translated into Medieval Latin in the 12th century by Gerard of Cremona, he used the Latin equivalent lasinusno (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a toga over the breast'). Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage. The English form sine was introduced in Thomas Fale's 1593 Horologiographia.

The word cosine derives from an abbreviation of the Latin 'sine of the complementary angle' as in Edmund Gunter's Canon triangulorum (1620), which also includes a similar definition of cotangens.

History

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).

The sine and cosine functions are closely related to the sajyā and sakoṭi-jyā functions used in Indian astronomy during the Gupta period (Aryabhatiya and Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.

In the early 17th-century, the French mathematician Albert Girard published the first use of the abbreviations sin, cos, and tan; these were further promulgated by Euler (see below). The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In a paper published in 1682, Leibniz proved that is not an algebraic function of . Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722). 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 the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.

Software implementations

There is no standard algorithm for calculating sine and cosine. IEEE 754, the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.

Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. sin(10).

A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or linearly interpolate between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.

The CORDIC algorithm is commonly used in scientific calculators.

The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to sin and cos.

Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.

In programming languages, sin and cos are typically either a built-in function (e.g. in Fortran and MATLAB) or found within the language's standard math library. For example, the C standard library defines sine functions within math.h: sin(double), sinf(float), and sinl(long double). The parameter of each is a floating point value, specifying the angle in radians. Each function returns the same data type as it accepts. Many other trigonometric functions are also defined in math.h, such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, Python defines math.sin(x) and math.cos(x) within the built-in math module. Complex sine and cosine functions are also available within the cmath module, e.g. cmath.sin(z). CPython's math functions call the C math library, and use a double-precision floating-point format.

Turns based implementations

Some software libraries provide implementations of sine and cosine using the input angle in half-turns radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases. These functions, following the convention set by Fortran, are typically called sinpi and cospi. For example, sinpi(x) is expressed in half-turns so that is the angle measure in radians. SciPy provides similar functions sindg and cosdg with input in degrees, as does Fortran (but named sind and cosd).

in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.

For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution. would be incurred. (See also Binary angular measurement.)