Sine and cosine

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Sine and cosine
Sine cosine one period.svg
General information
General definition
Fields of applicationTrigonometry, fourier series, etc.

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 that is 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. For an angle , the sine and cosine functions are denoted simply as and .[1]

More generally, the definitions of sine and cosine can be 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.

The functions sine and cosine can be traced to the functions jyā and koṭi-jyā, used in Indian astronomy during the Gupta period (Aryabhatiya and Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin.[2] The word sine (Latin sinus) comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, itself a transliteration of the Sanskrit word for half of a chord, jya-ardha.[3] The word cosine derives from a contraction of the medieval Latin complementi sinus.[4]

Notation[edit]

Sine and cosine are written using functional notation with the abbreviations sin and cos. Often if the argument is simple enough, the function value will be written without parentheses, as sin θ rather than as sin(θ). Each of sine and cosine is a function of an angle, which is usually expressed in terms of radians or degrees. Except where explicitly stated otherwise, this article assumes that the angle is measured in radians.

Definitions[edit]

Right-angled triangle definitions[edit]

For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.

To define the sine and cosine of an acute angle α, start with a right triangle that contains an angle of measure α; in the accompanying figure, angle α in triangle ABC is the angle of interest. The three sides of the triangle are named as follows:

  • The opposite side is the side opposite to the angle of interest, in this case side a.
  • The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
  • The adjacent side is the remaining side, in this case side b. It forms a side of (and is adjacent to) both the angle of interest (angle A) 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:[5]

The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides.[5]

As stated, the values and appear to depend on the choice of right triangle containing an angle of measure α. However, this is not the case: all such triangles are similar, and so the ratios are the same for each of them.

Unit circle definitions[edit]

In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.

Unit circle: a circle with radius one

Let a line through the origin intersect the unit circle, making an angle of θ with the positive half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos(θ) and sin(θ), respectively. This definition is consistent with the right-angled triangle definition of sine and cosine when 0 < θ < π/2: because the length of the hypotenuse of the unit circle is always 1, . The length of the opposite side of the triangle is simply the y-coordinate. A similar argument can be made for the cosine function to show that when 0 < θ < π/2, even under the new definition using the unit circle. tan(θ) is then defined as , or, equivalently, as the slope of the line segment.

Using the unit circle definition has the advantage that the angle can be extended to any real argument. This can also be achieved by requiring certain symmetries, and that sine be a periodic function.

Complex exponential function definitions[edit]

The exponential function is defined on the entire domain of the complex numbers. The definition of sine and cosine can be extended to all complex numbers via

These can be reversed to give Euler's formula

When plotted on the complex plane, the function for real values of traces out the unit circle in the complex plane.

When is a real number sine and cosine simplify to the imaginary and real parts of or , as:

When for real values and , sine and cosine can be expressed in terms of real sines, cosines, and hyperbolic functions as

Differential equation definition[edit]

is the solution to the two-dimensional system of differential equations and with the initial conditions and . One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions.

Series definitions[edit]

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.
This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.

The successive derivatives of sine, evaluated at zero, can be used to determine its Taylor series. 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. This means the successive derivatives of sin(x) are cos(x), -sin(x), -cos(x), sin(x), continuing to repeat those four functions. The (4n+k)-th derivative, evaluated at the point 0:

where the superscript represents repeated differentiation. This implies 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):[6]

Taking the derivative of each term gives the Taylor series for cosine:

Continued fraction definitions[edit]

The sine function can also be represented as a generalized continued fraction:

The continued fraction representations can be derived from Euler's continued fraction formula and express the real number values, both rational and irrational, of the sine and cosine functions.

Identities[edit]

Exact identities (using radians):

These apply for all values of .

Reciprocals[edit]

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. Similarly, the reciprocal of cosine is secant, which gives the ratio of the length of the hypotenuse to that of the adjacent side.

Inverses[edit]

The usual principal values of the arcsin(x) and arccos(x) functions graphed on the Cartesian plane

The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin−1). The inverse function of cosine is arccosine (arccos, acos, or cos−1). (The superscript of −1 in sin−1 and cos−1 denotes the inverse of a function, not exponentiation.) As sine and cosine are not injective, their inverses are not exact inverse functions, but partial inverse functions. For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain, the expression arcsin(x) will evaluate only to a single value, called its principal value. The standard range of principal values for arcsin is from π/2 to π and the standard range for arccos is from 0 to π.

where (for some integer k):

By definition, arcsin and arccos satisfy the equations:

and

Pythagorean trigonometric identity[edit]

The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:[1]

where sin2(x) means (sin(x))2.

Double angle formulas[edit]

Sine and cosine satisfy the following double angle formulas:

Sine function in blue and sine squared function in red. The X axis is in radians.

The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves. Specifically,[7]

The graph shows both the sine function and the sine squared function, with the sine in blue and 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.

Derivative and integrals[edit]

The derivatives of sine and cosine are:

and their antiderivatives are:

where C denotes the constant of integration.[1]

Properties relating to the quadrants[edit]

The four quadrants of a Cartesian coordinate system

The table below displays many of the key properties of the sine function (sign, monotonicity, convexity), arranged by the quadrant of the argument. For arguments outside those in the table, one may compute the corresponding information by using the periodicity of the sine function.

Quadrant Angle Sine Cosine
Degrees Radians Sign Monotony Convexity Sign Monotony Convexity
1st quadrant, I increasing concave decreasing concave
2nd quadrant, II decreasing concave decreasing convex
3rd quadrant, III decreasing convex increasing convex
4th quadrant, IV increasing convex increasing concave
The quadrants of the unit circle and of sin(x), using the Cartesian coordinate system

The following table gives basic information at the boundary of the quadrants.

Degrees Radians
Value Point type Value Point type
Root, inflection Maximum
Maximum Root, inflection
Root, inflection Minimum
Minimum Root, inflection

Fixed points[edit]

The fixed point iteration xn+1 = cos(xn) with initial value x0 = −1 converges to the Dottie number.

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 . The only real fixed point of the cosine function is called the Dottie number. That is, the Dottie number is the unique real root of the equation The decimal expansion of the Dottie number is .[8]

Arc length[edit]

The arc length of the sine curve between and is

where is the incomplete elliptic integral of the second kind with modulus . It cannot be expressed using elementary functions.

The arc length for a full period is[9]

where is the gamma function. This can also be written using and the lemniscate constant.[9][10]

Law of sines[edit]

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 proved 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.

Law of cosines[edit]

The law of cosines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

In the case where , and this becomes the Pythagorean theorem: for a right triangle, where c is the hypotenuse.

Special values[edit]

Some common angles (θ) shown on the unit circle. The angles are given in degrees and radians, together with the corresponding intersection point on the unit circle, (cos(θ), sin(θ)).

For certain integral numbers x of degrees, the values of sin(x) and cos(x) are particularly simple and can be expressed without nested square roots. A table of these angles is given below. For more complex angle expressions see Exact trigonometric values § Common angles.

Angle, x sin(x) cos(x)
Degrees Radians Gradians Turns Exact Decimal Exact Decimal
0 0g 0 0 0 1 1
15° 1/12π 16+2/3g 1/24 0.2588 0.9659
30° 1/6π 33+1/3g 1/12 1/2 0.5 0.8660
45° 1/4π 50g 1/8 0.7071 0.7071
60° 1/3π 66+2/3g 1/6 0.8660 1/2 0.5
75° 5/12π 83+1/3g 5/24 0.9659 0.2588
90° 1/2π 100g 1/4 1 1 0 0

90 degree increments:

x in degrees 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
sin x 0 1 0 −1 0
cos x 1 0 −1 0 1

Relationship to complex numbers[edit]

and are the real and imaginary parts of .

Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates (r, φ):

The real and imaginary parts are:

where r and φ represent the magnitude and angle of the complex number z.

For any real number θ, Euler's formula says that:

Therefore, if the polar coordinates of z are (r, φ),

Complex arguments[edit]

Domain coloring of sin(z) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.
sin(z) as a vector field

Applying the series definition of the sine and cosine to a complex argument, z, gives:

where sinh and cosh are the hyperbolic sine and cosine. These are entire functions.

It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:

Partial fraction and product expansions of complex sine[edit]

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

Alternatively, the infinite product for the sine can be proved using complex Fourier series.

Proof of the infinite product for the sine

Using complex Fourier series, the function can be decomposed as

Setting yields

Therefore, we get

The function is the derivative of . Furthermore, if , then the function such that the emerged series converges on some open and connected subset of is , which can be proved using the Weierstrass M-test. The interchange of the sum and derivative is justified by uniform convergence. It follows that

Exponentiating gives

Since and , we have . Hence

for some open and connected subset of . Let . Since converges uniformly on any closed disk, converges uniformly on any closed disk as well.[11] It follows that the infinite product is holomorphic on . By the identity theorem, the infinite product for the sine is valid for all , which completes the proof.

Usage of complex sine[edit]

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:

The complex sine function is also related to the level curves of pendulums.[how?][12][better source needed]

Complex graphs[edit]

Sine function in the complex plane
Complex sin real 01 Pengo.svg
Complex sin imag 01 Pengo.svg
Complex sin abs 01 Pengo.svg
real component imaginary component magnitude


Arcsine function in the complex plane
Complex arcsin real 01 Pengo.svg
Complex arcsin imag 01 Pengo.svg
Complex arcsin abs 01 Pengo.svg
real component imaginary component magnitude

History[edit]

Quadrant from 1840s Ottoman Turkey with axes for looking up the sine and versine of angles

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). See in particular Ptolemy's table of chords.

The function of sine and versine (1 − cosine) can be traced to the jyā and koṭi-jyā 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.[2]

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.[13] With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant.[13] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents.[14][15] 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°.[15]

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.[16] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722).[17] 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.[18]

Etymology[edit]

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 (جيب), which is however meaningless in that language and abbreviated jb (جب). Since Arabic is written without short vowels, jb was interpreted as the word jaib (جيب), which means 'bosom'. When the Arabic texts were translated in the 12th century into medieval Latin by Gerard of Cremona, he used the Latin equivalent for 'bosom', sinus (which also means 'bay' or 'fold').[19][20] 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.[21] The English form sine was introduced in the 1590s. The word cosine derives from a contraction of the Latin complementi sinus.[4]

Software implementations[edit]

There is no standard algorithm for calculating sine and cosine. IEEE 754-2008, the most widely used standard for floating-point computation, does not address calculating trigonometric functions such as sine.[22] 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(1022).

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.[citation needed]

The CORDIC algorithm is commonly used in scientific calculators.

The sine and cosine functions, along with other trigonometric functions, is 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 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[edit]

Some software libraries provide implementations of sine and cosine using the input angle in half-turns, a half-turn being an angle of 180 degrees or radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.[23][24] In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these function are called sinpi and cospi.[23][25][24][26][27][28] For example, sinpi(x) would evaluate to where x is expressed in radians.

The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing , , and 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.

Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo involves inaccuracies in representing .

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.[29] If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to would be incurred.

See also[edit]

Citations[edit]

  1. ^ a b c Weisstein, Eric W. "Sine". mathworld.wolfram.com. Retrieved 2020-08-29.
  2. ^ a b Uta C. Merzbach, Carl B. Boyer (2011), A History of Mathematics, Hoboken, N.J.: John Wiley & Sons, 3rd ed., p. 189.
  3. ^ 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.{{cite web}}: CS1 maint: archived copy as title (link)
  4. ^ a b "cosine".
  5. ^ a b "Sine, Cosine, Tangent". www.mathsisfun.com. Retrieved 2020-08-29.
  6. ^ See Ahlfors, pages 43–44.
  7. ^ "Sine-squared function". Retrieved August 9, 2019.
  8. ^ "OEIS A003957". oeis.org. Retrieved 2019-05-26.
  9. ^ a b "A105419 - Oeis".
  10. ^ Adlaj, Semjon (2012). "An Eloquent Formula for the Perimeter of an Ellipse" (PDF). American Mathematical Society. p. 1097.
  11. ^ Rudin, Walter (1987). Real and Complex Analysis (Third ed.). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 299, Theorem 15.4
  12. ^ "Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?". math.stackexchange.com. Retrieved 2019-08-12.
  13. ^ a b Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. p. 74. Archived from the original on 2013-10-19. Retrieved 2010-07-13.
  14. ^ Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media. ISBN 978-1-4020-0260-1.
  15. ^ a b "trigonometry". Encyclopedia Britannica.
  16. ^ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer. ISBN 9783540647676.
  17. ^ "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
  18. ^ See Merzbach, Boyer (2011).
  19. ^ Eli Maor (1998), Trigonometric Delights, Princeton: Princeton University Press, p. 35-36.
  20. ^ 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.{{cite web}}: CS1 maint: archived copy as title (link)
  21. ^ Smith, D.E. (1958) [1925], History of Mathematics, vol. I, Dover, p. 202, ISBN 0-486-20429-4
  22. ^ 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.{{cite web}}: CS1 maint: archived copy as title (link)
  23. ^ a b "MATLAB Documentation sinpi
  24. ^ a b "R Documentation sinpi
  25. ^ "OpenCL Documentation sinpi
  26. ^ "Julia Documentation sinpi
  27. ^ "CUDA Documentation sinpi
  28. ^ "ARM Documentation sinpi
  29. ^ "ALLEGRO Angle Sensor Datasheet

References[edit]

External links[edit]