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In this note we give an elementary derivation for the power series expansions for the sine and the cosine function.

We study the expected number of real roots of the random equation g1cosθ+g2cos2θ+…+gncosnθ=K where the coefficients gj's are normally distributed, but not necessarily all identical. It is shown that although this expected number is independent of the means of gj, (j=1,2,…,n), it will depend on their variances. The previous works in this direction considered the identical distribution for the coefficients.

We define and study multivariate sine and cosine functions, symmetric with respect to the alternating group An, which is a subgroup of the permutation (symmetric) group Sn. These functions are eigenfunctions of the Laplace operator. They determine Fourier-type transforms. There exist three types of such transforms: expansions into corresponding sine-Fourier and cosine-Fourier series, integral sine-Fourier and cosine-Fourier transforms, and multivariate finite sine and cosine transforms. In all these transforms, alternating multivariate sine and cosine functions are used as a kernel.

Questions involving the exact values of the trigonometric functions of 180, 360, 540 and 720 have been popular in compilations of challenging mathematical problems, and in the problems sections of mathematics journals. (See, for example, problems 189 and 339 in [1] and the sections entitled Trigonometry: numerical evaluation, Trigonometry: numerical identities and Trigonometry: theory of equations in [2].) Invariably, the determination of these values make little or no use of geometry; rather, the derivations rely heavily upon trigonometric identities, such as the double and triple angle formulas or the product-to-sum formulas. In this note, we will use a purely geometric approach to derive the values of the sine and cosine of multiples of 18?. Therefore, this capsule can be used as enrichment or investigative material for students in trigonometry or precalculus courses. Since 0? and 900 are already standard trigonometric angles, we consider only angles of measure 18?, 36?, 54? and 720. Using Figure 1, we first show that the sine and cosine of 18?, 36?, 540 and 72? can all be expressed as functions of x. Toward this end, draw the altitude CD from vertex C to the side AB (Figure(b)), and l n et BD = y. For the right triangle BCD, we find (by the Pythagorean theorem) that CD = x2 y2, and for the right triangle ADC, we see that CD = 1 (1 y)2 = 2y y2 Equating the expressions for CD leads to y = x2. Therefore

We prove several splitting theorems related to the following one: Every continuous real trigonometric rational function (of t, with period $2\pi $) can be written as the real part of a rational function of $z = e^{it} $ that is analytic for $|z| \leqq 1$. As an application, we obtain immediately a method for computing the conjugate function of a real trigonometric rational function.

A mechanical synthesizer with thirty harmonic elements (fifteen sine components and fifteen cosine components) may be used to graph a polynomial in a complex plane. The sum of the sine components is recorded by a tracing point which moves vertically. The sum of the cosine components is recorded by horizontal motion of a drawing board. Expansion of the polynomial by De Moivre's theorem expresses the function as a sum of sine terms and a sum of cosine terms. The polynomial may then be graphed by the machine, and thereby the complex roots determined. A number of typical curves are given to illustrate the machine method of determining real, imaginary, and complex roots. An auxiliary method is described which gives all the real roots of the polynomial from a single graph.

I. M. Vinogradov's method is used to derive estimates of partially rational trigonometric sums, independent of the degree of the rational part of the polynomial.

We consider double cosine and sine series whose coefficients form a null sequence of bounded variation. We prove that in this case the sum of a double cosine series is integrable in the sense of improper Riemann integral and the series in question is the Fourier series of its sum in the same sense. On the other hand, the sum of a double sine series is not necessarily integrable in the sense of improper Riemann integral, but the series is always a generalized Fourier sine series of its sum. These imply the important corollary that if the sum of a double cosine or sine series, with coefficients tending to zero and of bounded variation, is Lebesgue integrable, then the series is the Fourier series of its sum. Our results are the extensions of those by R. P. Boas and N. K. Bary from one-dimensional to two-dimensional trigonometric series.

(1987). The Derivatives of the Sine and Cosine Functions. The College Mathematics Journal: Vol. 18, No. 2, pp. 139-141.

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(1968). Sine and Cosine Integrals. Optica Acta: International Journal of Optics: Vol. 15, No. 2, pp. 195-196.

Were it not for Euclid’s Propositions iii. 26 (proved by superposition) and vi. 33, the statement that he nowhere defines the circumference of a circle or any curved line would surely be a truism. His definition of a circle is substantially only a locus definition. It may well be argued that neither does he define a straight line (or an angle) ; but in the case of a straight line he introduces sufficient ideas and axioms to make his subsequent propositions convincing, whereas he gives us nothing to bridge the interval between a straight line and a curved one.And so, wise after the event, we may say that all the thought spent in old days over ‘squaring the circle’ had no adequate foundation, and was foredoomed to fail.As to iii. 26 and vi. 33, they are only further examples of the truth with which we are familiar in the case of angles, that it is sometimes easier to prove that things are equal etc., than to say exactly what they are.