Cosine

Let $\theta$ be an Angle measured counterclockwise from the $x$-axis along the arc of the unit Circle. Then $\cos\theta$ is the horizontal coordinate of the arc endpoint. As a result of this definition, the cosine function is periodic with period $2\pi$.

The cosine function has a Fixed Point at 0.739085.

The cosine function can be defined algebraically using the infinite sum

$$\cos x\equiv \sum_{n=0}^\infty {(-1)^nx^{2n}\over (2n)!} = 1 - {x^2\over 2!} + {x^4\over 4!} - {x^6\over 6!} + \ldots,$$ (1)

or the Infinite Product

$$\cos x = \prod_{n=1}^\infty \left[{1 - {4x^2\over\pi^2(2n-1)^2}}\right].$$ (2)

A close approximation to $\cos(x)$ for $x\in [0,\pi/2]$ is

$$\cos\left({{\pi\over 2} x}\right)\approx 1-{x^2\over x+(1-x)\sqrt{2-x\over 3}}$$ (3)

(Hardy 1959). The difference between $\cos x$ and Hardy's approximation is plotted below.

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

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

$$= {\textstyle{1\over 2}}[\delta(k-k_0)+\delta(k+k_0)],$$ (4)

where $\delta(k)$ is the Delta Function.

The cosine sum rule gives an expansion of the Cosine function of a multiple Angle in terms of a sum of Powers of sines and cosines,

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

$$= \cos^n\theta-{n \choose 2}\cos^{n-2}\theta\sin^2\theta+{n\choose 4}\cos^{n-4}\theta\sin^4\theta-\ldots.$$ (5)

Summing the Cosine of a multiple angle from $n=0$ to $N-1$ can be done in closed form using

$$\sum_{n=0}^{N-1} \cos(nx) = \Re\left[{\,\sum_{n=0}^{N-1} e^{inx}}\right].$$ (6)

The Exponential Sum Formulas give

$$\sum_{n=0}^{N-1} \cos(nx) = \Re\left[{{\sin({\textstyle{1\over 2}}Nx)\over\sin({\textstyle{1\over 2}}x)} e^{i(N-1)x/2}}\right]$$

$$= {\sin({\textstyle{1\over 2}}Nx)\over\sin({\textstyle{1\over 2}}x)} \cos[{\textstyle{1\over 2}}x(N-1)].$$ (7)

Similarly,

$$\sum_{n=0}^\infty p^n \cos(nx) = \Re\left[{\,\sum_{n=0}^\infty p^ne^{inx}}\right],$$ (8)

where $\vert p\vert < 1$. The Exponential Sum Formula gives

$$\sum_{n=0}^\infty p^n \cos(nx) = \Re\left[{1-pe^{-ix}\over 1-2p\cos x+p^2}\right]$$

$$= {1-p\cos x\over 1-2p\cos x+p^2}.$$ (9)

Cvijovic and Klinowski (1995) note that the following series

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

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

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

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

See also Euler Polynomial, Exponential Sum Formulas, Fourier Transform--Cosine, Hyperbolic Cosine, Sine, Tangent, Trigonometric Functions

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.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 68, 1959.

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.