Angle, trigonometric functions

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 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... 

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. 

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] ... 

Directed Angles 27. Basic Trigonometric Functions 28. Radian Measure 28. Trigonometric Properties 29. Hyperbolic Functions 33. Polar Coordinate System 34. 

Trigonometric Function Values at Major Angle Values... 

Floiy (155) and other American writers use the opposite convention making 8 = 0° for the trans or anti conformation. As a result the numerical values of the torsion angles reported here are often different from those reported in the original literature. Differences are found in the formulas involving trigonometrical functions and in the appearance of energy maps. Specific conventions are used for vinyl polymers. 

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

Since negative angles arise when using trigonometric functions, it is important to establish how, for example, sin(-0) is related to sin0. The periodicity of the sine function yields the equality ... 

A prominent application of this rule is the case of a parametric dependence. In finding the derivative of trigonometric functions, the angle plays the role of a parameter and we can use Eq. (20) to calculate them. 

The coordinate map given by the variables (cD+, cD ) is a significant improvement as compared to eq. (3.25). Nevertheless, an explicit expression for an h matrix in its terms is still a clumsy combination of the trigonometric functions of two triples of reparametrizing angles w . It is known however that in the case of the SO(3) group [8] its quaternion [27] parameterization has the advantage that the matrix elements of SO(3) rotation matrices, when expressed in terms of the components of the normalized quaternion, are quadratic functions of these components. 

FVom this character table, and the symmetry eigenvectors of planar acetone (54-56), the symmetry eigenvectors of pyramidal acetone are easily deducible. For this purpose, linear combinations of the eigenvectors, which exhibit the same behavior for all the operations except for WU and VU, are built up. In addition, to the coefBcients of which are trigonometric functions of the wagging angle, a. The coefficients are chosen in such a way that the linear combinations fulfill the characters corresponding to operators WU and VU,... 

Figure 26.1 Right angled triangle used to define the trigonometric functions...

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