Trigonometric cosine

The frequency-dependent coefficients in this equation are given separate names and symbols to facilitate discussion. Remember it is these coefficients that determine the behavior of the system the trigonometric functions merely describe the oscillations. The following can be said of the coefficient of the cosine term ... 

The basic trigonometric functions are the sine, cosine, and tangent. 

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

Similarly, many different types of functions can be used. Arden discusses, for example, the use of Chebyshev polynomials, which are based on trigonometric functions (sines and cosines). But these polynomials have a major limitation they require the data to be collected at uniform -intervals throughout the range of X, and real data will seldom meet that criterion. Therefore, since they are also by far the simplest to deal with, the most widely used approximating functions are simple polynomials they are also convenient in that they are the direct result of applying Taylor s theorem, since Taylor s theorem produces a description of a polynomial that estimates the function being reproduced. Also, as we shall see, they lead to a procedure that can be applied to data having any distribution of the X-values. 

Working independently, A.Abakonovicz in 1878 and C.V. Boys in 1882 devised the integraph, an instrument that drew the integral of an arbitrary function when the latter was plotted on a suitable scale on paper. A device for finding trigonometric functions (sines and cosines), known as harmonic analyzer was devised in 1876 by Lord Kelvin. 

Cosine trigonometric functions, in other words, are given by the real part of the function e10. This means that Equations (80) and (83) may be written... 

The relatively simple inversion of the direction-cosine matrix can be seen from trigonometric identities among direction cosines, stated as... 

Consider the right-angled triangle shown in Figure 2.13. The basic trigonometric functions sint and cosine, given the names sin and cos,... 

We can see from Table 2.5 and Figure 2.17 that the sine and cosine functions both have as domain the set of real numbers. The domains of the tangent and reciprocal trigonometric functions are different, however,... 

In words, the difference Patterson function is a Fourier series of simple sine and cosine terms. (Remember that the exponential term is shorthand for these trigonometric functions.) Each term in the series is derived from one reflection hkl in both the native and derivative data sets, and the amplitude of each term is (IFHpI — IFpl)2, which is the amplitude contribution of the heavy atom to structure factor FHp. Each term has three frequencies h in the u-direction, k in the v-direction, and l in the w-direction. Phases of the structure factors are not included at this point, they are unknown. 

There are special ratios we can use when working with right triangles. They are based on the trigonometric functions called sine, cosine, and tangent. 

As a second example, suppose that the crystal has three mutually perpendicular, two-fold axes of symmetry. In this case, if we take these axes as the axes of coordinates, each term in the Fourier series may contain the product of three cosines, but, unless the coordinate planes are also planes of symmetry, the series must contain terms with trigonometric sines. These sines, however, must occur in pairs. A term cannot contain the product of one sine and two cosines, nor can it be the product of three sines for the term must have the same value when we change the algebraic signs... 

Note that the 180 -y pulse on the 13C channel has no effect on Sv. The cosine term is just the product operator we started with, unaffected by the1H pulse, and the sine term is the operator we would get with a full 90° 1H pulse. Note that rotation of the lx magnetization vector by a XH B field on the / axis goes from x to — z to — x to +z as is incremented from 0° to 90° to 180° to 270° in the trigonometric expression. The first term is DQC/ZQC, which will not be observable in the FID—there are no more pulses in the sequence to convert it to observable magnetization. Only the second term represents full coherence transfer to antiphase 13C coherence, which will refocus during the final 1/(27) delay into in-phase 13C coherence ... 

The integrals I0 and /2 arise from a Fourier expansion for hyperbolic cosines of trigonometric argument [45]. With this potential, Parker calculated both the average number of collisions to establish rotational equilibrium, Zf, and that required to establish vibrational equilibrium, Z , as functions of temperature. One finds... 

If we develop the torsional solutions of equation (113) on the basis of double products of trigonometric functions (built up with 16 cosine and 15 sine ones), the Hamiltonian matrix to be diagonalized would be of order 31 x 31 = 961. 

The three trigonometric functions sine, cosine and tangent are then defined by the equations ... 

Other functions commonly differentiated in chemistry are the sine and cosine trigonometric functions. The relevant derivatives are ... 

A function may be represented by a graph, which is a picture of how the value of the function (dependent variable) changes when the independent variable changes. Some of the more common periodic functions include the sawtooth, the square wave, and the trigonometric functions (sine, cosine, and tangent) (Figure 1). 

Any periodic function (such as the electron density in a crystal which repeats from unit cell to unit cell) can be represented as the sum of cosine (and sine) functions of appropriate amplitudes, phases, and periodicities (frequencies). This theorem was introduced in 1807 by Baron Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist who pioneered, as a result of his interest in a mathematical theory of heat conduction, the representation of periodic functions by trigonometric series. Fourier showed that a continuous periodic function can be described in terms of the simpler component cosine (or sine) functions (a Fourier series). A Fourier analysis is the mathematical process of dissecting a periodic function into its simpler component cosine waves, thus showing how the periodic function might have been been put together. A simple... 

The product of the signals can be expanded using the trigonometric identity for the cosine of the sum of two angles and integrated over each cycle. Only the leading term of the series has a nonzero value, thus. 

Here p means either of the trigonometric functions sine or cosine. When p represents sine,... 

Application of trigonometric identities allows a product of cosines to be re-written as a sum ... 

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