Trigonometric expressions used

Figure 3. Trigonometric expressions used in calculation of dwell. The helical wind angle is A. ...

Structural dependences of 4J(F1,F[) in propanic and allylic systems were recently studied by Barfield29 calculating, within the FPT-DFT approach, the FC term of such couplings. In propane Barfield carried out calculations for different values of the dihedral angles about the C3 C2 and C2 C3 bonds and for different values of the C3 C2 C3 internal angle. Using these calculated values he established a four-term trigonometric expression, which was then... 

In semiempirical methods, each orbital on an atom has an unique angular function. These functions can be expressed using either Cartesian coordinates or trigonometric functions. The set of normalized functions most commonly used is given in Table 2, in which 9 is the polar angle from the z axis, and

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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] ... 

This formula yields the angle A expressed in degrees, which requires the use of a trigonometric table or a calculator that is capable of determining the inverse tangent. 

Trigonometric substitutions are often useful in evaluating integrals. Among the many possibilities, if the integrand involves the expression x2 + a2, the substitution x = a tamp should be tried. Similarly, in the cases of x2 - a2 or a2 - x2, the independent variable x should be replaced by a sec ip ova sin [Pg.238]

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave ... 

It often happens that the coordinates are not transformed simply into each other by a symmetry operation. Trigonometric relations must be used to express, for instance, the consequences of three-fold rotation. 

We can picture the peaks more easily if we use trigonometric identities to obtain for expressions (3) and (4), that is, one diagonal and one cross peak, the following ... 

Since Equation (3) is nonlinear and transcendental i.e. invokes trigonometric functions), the usual linear version of least-squares analysis can not be used. The method is to approximate the expression for as a Taylor series and only retain the first term ... 

It was noted above that shifting the phase of the first pulse in the NOESY sequence from x to -y caused the modulation to change from cos Qtx) to sin j). One way of expressing this is to say that shifting the pulse causes a phase shift (j) in the signal modulation, which can be written cos + (pi). Using the usual trigonometric expansions this can be written... 

Unlike the expression just derived, the relationship the problem asks us to derive has no trigonometric functions and it contains E and a within a square root. This comparison suggests that the trigonometric identity sin2 6 + cos2 6 = 1 will be of use here. Let 8 = kji/(N + 1) then... 

Recently Wake has applied a variational treatmrait to the stationary problem, deriving critical conditions both for the class A geometries and for the cube, square rod, and equicylinder in systems where the heat transf(H is resisted by conduction in the interior and by convection at the surface. Here the condition at the boundary becomes dO/dp + N6 = 0, where N is the Biot number hLIk The limit as bf- oo corresponds to the Frank-Kamenetskii solutions. Wake uses trigonometric, rather than polynomial, expressions for this tempoature field and proceeds to derive the conditions under which solutions of the time-dependent variational equations are just possible, associating these with a critical value of 6. Results for N = oo are listed as variational (2) in the Table. For the more rorai conditions of finite Biot numbers Wake compares his results for class A geometries with the analytical forms due to Thomas. Errors are less than 0.1 % though the computational effort required is substantial. 

This notation simplifies considerably the mathematical expressions, and there is no need to use complicated trigonometric formulae when dealing with the changes in the phase and ampUtude of AC signals. 

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