نطقب:Infobox mathematics function
- تحويل نطقب:حساب المثلثات
جيب Sine في الرياضيات هوالنسبة بين الضلع اللقاء لزاوية والوتر في مثلث ذوزاوية قائمة ، بحيثقد يكون الوتر هوالضلع اللقاء للزاوية القائمة.
في رياضيات، تعتبر التوابع مثلثية أوالدوال المثلثية دوال لزاوية هندسية، وهي دوال مهمة عندما نريد دراسة مثلث أوعرض ظواهرِ دورية. يمكن تعريف هذه الدوال كنسبة لأضلاع مثلث قائم الذي يَحتوي تلك الزاويةَ أَوبشكل أكثر عمومية كإحداثيات على دائرة مثلثية أودائرة واحدية (unit circle) . في الرياضيات ، الدوال المثلثية هي دوال ترتبط بالزاوية، وهي مهمة في دراسة المثلثات وتمثيل الظواهر المتكررة (كالموجات). ويمكن تعريف الدوال المثلثية على أنهم نسب بين ضلعين في مثلث قائم فيه الزاوية المعنية، او، وبشكل أوسع. كنسبة بين إحداثيات نقاط على دائرة الوحدة، ويعتبر دوما عند الإشارة إلى المثلثات ان الحديث يدور حول مثلث في سطح مستوي (مستوى إحداثي أوإقليدي) ، وذلك ليكون مجموع الزوايا 180 درجة دائما.
وهناك ثلاثة دوال مثلثية أساسية هي:
- جا أوالجيب ، ويساوي النسبة بين الضلع اللقاء للزاوية مقسوما على الوتر.
- جتا أوجيب التمام ، ويساوي النسبة بين الضلع المجاور للزاوية مقسوما على الوتر.
- ظا اوالظل ، ويساوي النسبية بين الضلع اللقاء للزاوية والضلع المجاور لها.
Exact identities (using radians):
These apply for all values of .
The reciprocal of sine is cosecant, i.e., the reciprocal of sin(A) is csc(A), or cosec(A). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:
For the sine function:
The derivative is:
The antiderivative is:
C denotes the constant of integration.
Other trigonometric functions
It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function).
Sine in terms of the other common trigonometric functions:
|f θ||Using plus/minus (±)||Using sign function (sgn)|
|f θ =||± per Quadrant||f θ =|
Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant.
The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity:
where sin2x means (sin(x))2.
Properties relating to the quadrants
Over the four quadrants of the sine function is as follows.
Points between the quadrants. k is an integer.
For arguments outside those in the table, get the value using the fact the sine function has a period of 360° (or 2π rad): , or use . Or use and . For complement of sine, we have .
Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of sine.
Using the reflection from the calculated geometric derivation of the sine is with the 4n + k-th derivative at the point 0:
This gives the following Taylor series expansion at x = 0. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x (where x is the angle in radians) :
If x were expressed in degrees then the series would contain factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx /180, so
The series formulas for the sine and cosine are uniquely determined, up to the choice of unit for angles, by the requirements that
The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that
The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients.
In general, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.
A similar series is Gregory's series for arctan, which is obtained by omitting the factorials in the denominator.
The sine function can also be represented as a generalized continued fraction:
The continued fraction representation can be derived from Euler's continued fraction formula and expresses the real number values, both rational and irrational, of the sine function.
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 is sin(0) = 0.
The arc length of the sine curve between and is This integral is an elliptic integral of the second kind.
The arc length for a full period is where
The arc length of the sine curve from 0 to x is the above number divided by times x, plus a correction that varies periodically in x with period
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:
This is equivalent to the equality of the first three expressions below:
where R is the triangle's circumradius.
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.
For certain integral numbers x of degrees, the value of sin(x) is particularly simple. A table of some of these values is given below.
|x (angle)||sin x|
90 degree increments:
|x in degrees||0°||90°||180°||270°||360°|
|x in radians||0||π/2||π||3π/2||2π|
|x in gons||0||100g||200g||300g||400g|
|x in turns||0||1/4||1/2||3/4||1|
Other values not listed above:
Relationship to complex numbers
Sine is used to determine the imaginary part of a complex number given in polar coordinates (r,φ):
the imaginary part is:
r and φ represent the magnitude and angle of the complex number respectively. i is the imaginary unit. z is a complex number.
Although dealing with complex numbers, sine's parameter in this usage is still a real number. Sine can also take a complex number as an argument.
Sine with a complex argument
The definition of the sine function for complex arguments z:
where i 2 = −1, and sinh is hyperbolic sine. This is an entire function. Also, for purely real x,
For purely imaginary numbers:
It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument:
Partial fraction and product expansions of complex sine
Using the partial fraction expansion technique in complex analysis, one can find that the infinite series
both converge and are equal to . Similarly, one can show that
Using product expansion technique, one can derive
Usage of complex sine
sin z 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, sin z is a 2D solution of Laplace's equation:
It is also related with level curves of pendulum.
|real component||imaginary component||magnitude|
|real component||imaginary component||magnitude|
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 function sine (and cosine) can be traced to the functions used in Gupta period (320 to 550 CE) Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.
The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician Albert Girard; 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 sin x is not an algebraic function of x.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.
Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym). This was transliterated in Arabic as jiba جــيــب, which however is meaningless in that language and abbreviated jb جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جــيــب, which means "bosom". When the Arabic texts were translated in the 12th century into Latin by Gerard of Cremona, he used the Latin equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold"). 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 the 1590s.
The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, it is typically abbreviated to
Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.
In programming languages,
sin is typically either a built-in function or found within the language's standard math library.
For example, the C standard library defines sine functions within math.h:
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) within the built-in
math module. Complex sine 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.
There is no standard algorithm for calculating sine. IEEE 754-2008, the most widely used standard for floating-point computation, does not address calculating trigonometric functions such as sine. 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.
A once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sine values, for example one value per degree. This allowed 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.
- Āryabhaṭa's sine table
- Bhaskara I's sine approximation formula
- Discrete sine transform
- Euler's formula
- Generalized trigonometry
- Hyperbolic function
- Law of sines
- List of periodic functions
- List of trigonometric identities
- Madhava series
- Madhava's sine table
- Optical sine theorem
- Polar sine — a generalization to vertex angles
- Proofs of trigonometric identities
- Sinc function
- Sine and cosine transforms
- Sine integral
- Sine quadrant
- Sine wave
- Sine–Gordon equation
- Sinusoidal model
- Trigonometric functions
- ^ See Ahlfors, pages 43–44.
- ^ math.stackexchange questions : why-are-the-phase-portrait-of-the-simple-plane-pendulum-and-a-domain-coloring-of ... Archived 2014-03-30 at the Wayback Machine.
^ خطأ استشهاد: وسم
<ref>غير سليم؛ لا نص تم توفيره للمراجع المسماة
Boyer, Carl B. 1991 p. 210
- ^ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer.
- ^ "Why the sine has a simple derivative Archived 2011-07-20 at the Wayback Machine.", in Historical Notes for Calculus Teachers Archived 2011-07-20 at the Wayback Machine. by V. Frederick Rickey Archived 2011-07-20 at the Wayback Machine.
- ^ See Merzbach, Boyer (2011).
- ^ Eli Maor (1998), Trigonometric Delights, Princeton: Princeton University Press, p. 35-36.
- ^ Victor J. Katz (2008), A History of Mathematics, Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. "Archived copy" (PDF). Archived (PDF) from the original on 2015-04-14. Retrieved 2015-04-09. CS1 maint: archived copy as title (link)
- ^ Smith, D.E. (1958) , History of Mathematics, I, Dover, p. 202, ISBN 0-486-20429-4
- ^ Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31 "Archived copy" (PDF). Archived (PDF) from the original on 2011-07-16. Retrieved 2010-09-11. CS1 maint: archived copy as title (link)
- Traupman, John C. (1966), The New College Latin & English Dictionary, Toronto: Bantam, ISBN 0-553-27619-0
- Webster's Seventh New Collegiate Dictionary, Springfield: G. & C. Merriam Company, 1969
- Media related to Sine function at Wikimedia Commons