Trigonometric

A. Zigmund, Trigonometrical Series, Dover, New York, 1955, Chap. III. 

W. Gautschi. Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3 381-397, 1961. 

Series for the Trigonometric Functions. In the following formulas, all angles must be expressed in radians. If D = the number of degrees in the angle, and x = its radian measure, then x = 0.017453Z7. 

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 ... 

Describe the angular dependence of the vertically and horizontally polarized light scattered by a molecule and their resultant by considering the intensity as a vector anchored at the origin whose length in various directions is given by the trigonometric terms in Eqs. (10.25), (10.26), and (10.30),... 

The trigonometric functions of angles are the ratios between the various sides of the reference triangles shown in Fig. 3-39 for the various quadrants. Clearly r = /x + y > 0. The fundamental functions (see Figs. 3-40, 3-41, 3-42) are... 

Trigonometric Substitution This technique is particularly well adapted to integrands in the form of radicals. For these the function is transformed into a trigonometric form. In the latter form they may be more easily recognizable relative to the identity formulas. These functions and their transformations are... 

No general rule for breaking an integrand can be given. Experience alone limits the use of this technique. It is particularly useful for trigonometric and exponential functions. 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

If DV < 0, the roots are real and unequal and a trigonometric solution is preferred. 

Tbiecalculatorlevelof Mathematica comesintbieformofPalettes,wbiicbiareverybiandytools. Palettes are found under the File menu and there are several of them. If one wants to use a trigonometric function, for example, we can either type in its name or go to the Basic... 

Calculations menu and then to the Trigonometric and Exponential Functions. Should we wanttoevaluatethesineof2.33337T, thenwecandosoasfollows ... 

Tsai and Pagano [2-7] ingeniously recast the stiffness transformation equations to enable ready understanding of the consequences of rotating a lamina in a laminate. By use of various trigonometric identities between sin and cos to powers and sin and cos of multiples of the angle, the transformed reduced stiffnesses. Equation (2.85), can be written as... 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

Using trigonometric identities, the off-diagonal elements may be written as... 

The transformation from a set of Cartesian coordinates to a set of internal coordinates, wluch may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-linear transformation will affect the convergence properties. This may be illustrate by considering a minimization of a Morse type function (eq. (2.5)) with D = a = ] and x = AR. 

Triftholz, n. driftwood, triftig, a. weighty, sound, valid adrift. Triftiohr, n. (Phyaica) klystron, trigonometrisch, a. trigonometric.. 

The correct determination of/ depends largely on using the correct value of the eios angle. In light of the analysis of a coauthor [19], the value of 105 is equal to 105 35.8 . Assuming this 105 value and the occurrence of the unit cell proposed by Daubeny and Bunn [8], after calculation of values of trigonometric functions, the expression in Eq. (6) may be written in the form [19] ... 

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