Sine

Let $\theta$ be an Angle measured counterclockwise from the x-Axis along the arc of the Unit Circle. Then $\sin\theta$ is the vertical coordinate of the arc endpoint. As a result of this definition, the sine function is periodic with period $2\pi$. By the Pythagorean Theorem, $\sin\theta$ also obeys the identity

$$\sin^2\theta+\cos^2\theta=1.$$ (1)

The sine function can be defined algebraically by the infinite sum

$$\sin x=\sum_{n=1}^\infty {(-1)^{n-1}\over (2n-1)!}x^{2n-1}$$ (2)

and Infinite Product

$$\sin x = x \prod_{n=1}^\infty \left({1 - {x^2\over n^2\pi^2}}\right).$$ (3)

It is also given by the Imaginary Part of the complex exponential

$$\sin x=\Im[e^{ix}].$$ (4)

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

$$\csc x\equiv {1\over\sin x}.$$ (5)

Using the results from the Exponential Sum Formulas

$$\sum_{n=0}^N \sin(nx) = \Im\left[{\,\sum_{n=0}^N e^{inx}}\right]$$

$$= \Im\left[{{\sin({\textstyle{1\over 2}}Nx)\over\sin({\textstyle{1\over 2}}x)} e^{i(N+1)x/2}}\right]$$

$$= {\sin({\textstyle{1\over 2}}Nx)\over\sin({\textstyle{1\over 2}}x)} \sin[{\textstyle{1\over 2}}x(N+1)].$$ (6)

Similarly,

$$\sum_{n=0}^\infty p^n\sin(nx) = \Im\left[{\,\sum_{n=0}^\inft... ...x}\over 1-2p\cos x+p^2}\right]= {p\sin x\over 1-2p\cos x+p^2}.$$ (7)

Other identities include

$$\sin(n\theta)=2\cos\theta\sin[(n-1)\theta]-\sin[(n-2)\theta]$$ (8)

$$\sin(nx) = {n\choose 1}\cos^{n-1}x\sin x - {n\choose 3}\cos^{n-3}x\sin^3 x + {n\choose 5}\cos^{n-5}x\sin^5 x - \ldots,$$ (9)

where ${n\choose k}$ is a Binomial Coefficient.

Cvijovic and Klinowski (1995) show that the sum

$$S_\nu(\alpha)=\sum_{k=0}^\infty {\sin(2k+1)\alpha\over(2k+1)^\nu}$$ (10)

has closed form for $\nu=2n+1$,

$$S_{2n+1}(\alpha)={(-1)^n\over 4(2n)!}\pi^{2n+1}E_{2n}\left({\alpha\over\pi}\right),$$ (11)

where $E_n(x)$ is an Euler Polynomial.

A Continued Fraction representation of $\sin x$ is

$$\sin x={x\over 1+{\strut\displaystyle x^2\over\strut\display... ...4\cdot 5x^2\over\strut\displaystyle (6\cdot 7-x^2)+\ldots}}}}.$$ (12)

The value of $\sin(2\pi/n)$ is Irrational for all $n$ except 4 and 12, for which $\sin(\pi/2)=1$ and $\sin(\pi/6)=1/2$.

The Fourier Transform of $\sin(2\pi k_0x)$ is given by

$${\mathcal F}[\sin(2\pi k_0x)] = \int_{-\infty}^\infty e^{-2\pi ikx}\sin(2\pi k_0x)\,dx$$

$$= {\textstyle{1\over 2}}i[\delta(k+k_0)-\delta(k-k_0)].$$ (13)

Definite integrals involving $\sin x$ include

$$\int_0^\infty \sin(x^2)\,dx = {\textstyle{1\over 4}}\sqrt{2\pi}$$ (14)

$$\int_0^\infty \sin(x^3)\,dx = {\textstyle{1\over 6}}\Gamma({\textstyle{1\over 3}})$$ (15)

$$\int_0^\infty \sin(x^4)\,dx = -\cos({\textstyle{5\over 8}}\pi)\Gamma({\textstyle{5\over 4}})$$ (16)

$$\int_0^\infty \sin(x^5)\,dx = {\textstyle{1\over 4}}(\sqrt{5}-1)\Gamma({\textstyle{6\over 5}}),$$ (17)

where $\Gamma(x)$ is the Gamma Function.

See also Andrew's Sine, Cosecant, Cosine, Fourier Transform--Sine, Hyperbolic Sine, Sinc Function, Tangent, Trigonometry

References

Abramowitz, M. and Stegun, C. 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.

Cvijovic, D. and Klinowski, J. ``Closed-Form Summation of Some Trigonometric Series.'' Math. Comput. 64, 205-210, 1995.

Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975.

Project Mathematics! Sines and Cosines, Parts I-III. Videotapes (28, 30, and 30 minutes). California Institute of Technology. Available from the Math. Assoc. Amer.

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.