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Sine

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Trigonometry
Sin

The sine function sinx is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let theta be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then sintheta is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.

SineDiagram

The common schoolbook definition of the sine of an angle theta in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e.,

sintheta=(opposite)/(hypotenuse).
(1)

A convenient mnemonic for remembering the definition of the sine, as well as the cosine and tangent, is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).

As a result of its definition, the sine function is periodic with period 2pi. By the Pythagorean theorem, sintheta also obeys the identity

sin^2theta+cos^2theta=1.
(2)
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The definition of the sine function can be extended to complex arguments z, illustrated above, using the definition

sinz=(e^(iz)-e^(-iz))/(2i)
(3)
=1/2i(e^(-iz)-e^(iz)),
(4)

where e is the base of the natural logarithm and i is the imaginary number. Sine is an entire function and is implemented in the Wolfram Language as Sin[z].

A related function known as the hyperbolic sine is similarly defined,

sinhz=1/2(e^z-e^(-z)).
(5)

The sine function can be defined analytically by the infinite sum

sinx=sum_(n=1)^infty((-1)^(n-1))/((2n-1)!)x^(2n-1).
(6)

It is also given by the imaginary part of the complex exponential

sinx=I[e^(ix)].
(7)

The multiplicative inverse of the sine function is the cosecant, defined as

cscx=1/(sinx).
(8)

The sine function is also given by the limit

sin(z)=-pilim_(n->infty)1/(lnn)sum_(k=1)^infty(mu(k))/kln(n/k)frac((kz)/(2pi)),
(9)

where mu(k) is the Möbius function and frac(x) is the fractional part (M. Trott).

The derivative of sinx is

d/(dx)sinx=cosx,
(10)

and its indefinite integral is

intsinxdx=-cosx+C,
(11)

where C is a constant of integration.

Using the results from the exponential sum formulas

sum_(n=0)^(N)sin(nx)=I[sum_(n=0)^(N)e^(inx)]
(12)
=I[(e^(i(N+1)x)-1)/(e^(ix)-1)]
(13)
=I[(e^(i(N+1)x/2))/(e^(ix/2))(e^(i(N+1)x/2)-e^(-i(N+1)x/2))/(e^(ix/2)-e^(-ix/2))]
(14)
=(sin(1/2(N+1)x))/(sin(1/2x))I[e^(iNx/2)]
(15)
=(sin(1/2Nx)sin[1/2(N+1)x])/(sin(1/2x)).
(16)

Similarly,

sum_(n=0)^(infty)p^nsin(nx)=I[sum_(n=0)^(infty)p^ne^(inx)]
(17)
=I[(1-pe^(-ix))/(1-2pcosx+p^2)]
(18)
=(psinx)/(1-2pcosx+p^2).
(19)

The sum of sin^2(kx) can also be done in closed form,

sum_(k=0)^Nsin^2(kx)=1/4{1+2N-cscxsin[x(1+2N)]}.
(20)

A related sum identity is given by

sum_(k=1)^(n-1)sin((kpi)/n)=cot(pi/(2n))
(21)

(T. Drane, pers. comm., Apr. 19, 2006).

Product identities include

pisproduct_(n=1)^infty(1-(s^2)/(n^2))=sin(pis),
(22)

which is more commonly written as an identity for the sinc function or in the form

sinx=xproduct_(n=1)^infty(1-(x^2)/(n^2pi^2))
(23)

(Edwards 2001, pp. 18 and 47; Borwein et al. 2004, p. 5). Another product formula is

product_(k=1)^(n-1)sin((kpi)/n)=2^(1-n)n
(24)

(T. Drane, pers. comm., Apr. 19, 2006).

The sine function obeys the identity

sin(ntheta)=2costhetasin[(n-1)theta]-sin[(n-2)theta]
(25)

and the multiple-angle formula

sin(nx)=sum_(k=0)^n(n; k)cos^kxsin^(n-k)xsin[1/2(n-k)pi],
(26)

where (n; k) is a binomial coefficient. It is related to tan(x/2) via

sinx=(2tan(1/2x))/(1+tan^2(1/2x))
(27)

(Trott 2006, p. 39).

A curious identity is given by

(sin(nalpha))/(sinalpha)=sum_(j=1)^nproduct_(k=1; k!=j)^n(sin(alpha+theta_j-theta_k))/(sin(theta_j-theta_k))
(28)

for all alpha and theta_j!=theta_k (Calogero 1999; Beylkin and Mohlenkamp 2002; Trott 2005, pp. 5-6).

Cvijović and Klinowski (1995) show that the sum

S_nu(alpha)=sum_(k=0)^infty(sin(2k+1)alpha)/((2k+1)^nu)
(29)

has closed form for nu=2n+1,

S_(2n+1)(alpha)=((-1)^n)/(4(2n)!)pi^(2n+1)E_(2n)(alpha/pi),
(30)

where E_n(x) is an Euler polynomial.

A continued fraction representation of sinx is

sinx=x/(1+(x^2)/((2·3-x^2)+(2·3x^2)/((4·5-x^2)+(4·5x^2)/((6·7-x^2)+...))))
(31)

(Olds 1963, p. 138).

The value of sin(2pi/n) is irrational for all integers n>1 except 2, 4, and 12, for which sin(pi)=0, sin(pi/2)=1, and sin(pi/6)=1/2, respectively, a result that is essentially known as Niven's theorem.

The Fourier transform of sin(2pik_0x) is given by

F_x[sin(2pik_0x)](k)=int_(-infty)^inftye^(-2piikx)sin(2pik_0x)dx
(32)
=1/2i[delta(k+k_0)-delta(k-k_0)].
(33)

A definite integral involving sinx is given by

int_0^inftysin(x^n)dx=Gamma(1+1/n)sin(pi/(2n))
(34)

for n>1 where Gamma(z) is the gamma function (R. Mabry, pers. comm., Dec. 15, 2005; T. Drane, pers. comm., Apr. 21, 2006).

See also

Andrew's Sine, Cis, Cosecant, Cosine, Elementary Function, Fourier Transform--Sine, Hyperbolic Polar Sine, Hyperbolic Sine, Hypersine, Inverse Sine, Niven's Theorem, Polar Sine, Sinc Function, Sinusoid, SOHCAHTOA, Tangent, Trigonometric Functions, Trigonometry Explore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/ElementaryFunctions/Sin/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Circular Functions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972.Beylkin, G. and Mohlenkamp, M. J. Proc. Nat. Acad. Sci. USA 99, 10246, 2002.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Calogero, F. "Remarkable Matrices and Trigonometric Identities. II." Commun. Appl. Math. 3, 267-270, 1999.Cvijović, D. and Klinowski, J. "Closed-Form Summation of Some Trigonometric Series." Math. Comput. 64, 205-210, 1995.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975.Olds, C. D. Continued Fractions. New York: Random House, 1963.Project Mathematics. "Sines and Cosines, Parts I-III." Videotape. http://www.projectmathematics.com/sincos1.htm.Jeffrey, A. "Trigonometric Identities." §2.4 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 111-117, 2000.Spanier, J. and Oldham, K. B. "The Sine sin(x) and Cosine cos(x) Functions." Ch. 32 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 295-310, 1987.Tropfke, J. Teil IB, §1. "Die Begriffe des Sinus und Kosinus eines Winkels." In Geschichte der Elementar-Mathematik in systematischer Darstellung mit besonderer Berücksichtigung der Fachwörter, fünfter Band, zweite aufl. Berlin and Leipzig, Germany: de Gruyter, pp. 11-23, 1923.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. http://www.mathematicaguidebooks.org/.Zwillinger, D. (Ed.). "Trigonometric or Circular Functions." §6.1 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 452-460, 1995.

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Sine

Cite this as:

Weisstein, Eric W. "Sine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Sine.html

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