Trigonometric functions of an angle

Trigonometrical ratios of an angle as functions of the sides of a triangle. There are certain functions of the angles, or rather of the arc PA (Fig. 181)... 

With reference to Fig. 23.41, P (x, y) is a point in any one of the four quadrants and A is an angle whose initial side is coincident with the positive x axis and whose terminal side contains the point P (x, y). The distance from the origin P(x,y) is denoted by r and is positive. The trigonometric functions of the angle... 

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. 

However, the expansion above is still impractical for our purposes, because the functions As(q), Bs(q),. .. still need to be expanded in an infinite Fourier series of the angles e.g., we should write A q) = J2kez akexp(i(k,qj). It is more convenient to work with trigonometric polynomials, so that every part of the expansion contains only a finite number of terms. To this end, we introduce a Fourier cutoff by splitting every function of the angles in an infinite number of slices that contain only a finite number of Fourier modes. This may be done in many arbitrary ways, so let us illustrate just one method. We choose an arbitray integer K, ad write, e.g.,... 

Trigonometric functions are examples of mathematicai functions To a given value of an angle there corresponds a value of the sine function, and so on. 

This is called the circular measure of an angle and, for this reason, trigonometrical functions are sometimes called circular functions. This property is possessed by no plane curve other than the circle. For instance, the hyperbola, though symmetrically placed with respect to its centre, is not at all points equidistant from it. The same thing is true of the ellipse. The parabola has no centre. 

MATLAB offers a large selection of built-in functions that you can use to analyze data. As we discussed in the previous chapter, by built-in funaions we mean standard functions such as the sine or cosine of an angle, as well as formulas that calculate the total value, the average value, or the standard deviation of a set of data points. The MATLAB functions are available in various categories, including mathematical, trigonometric, statistical, and logical functions. In this chapter, we will discuss some of the common functions. MATLAB offers a Help menu that you can use to obtain information on various commands and functions. 

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. 

Spreadsheets have program-specific sets of predetermined functions but they almost all include trigonometrical functions, angle functions, logarithms (p. 262) and random number functions. Functions are invaluable for transforming sets of data rapidly and can be used in formulae required for more complex analyses. Spreadsheets work with an order of preference of the operators in much the same way as a standard calculator and this must always be taken into account when operators are used in formulae. They also require a very precise syntax - the program should warn you if you break this ... 

A disadvantage of the Euler angle approach is that the rotation matrix contains a total of six trigonometric functions (sine and cosine for each of the three Euler angles). These trigonometric functions are computationally expensive to calculate. An alternative is to use quaternions. A quaternion is a four-dimensional vector such that its components sum to 1 0 + 1 + <72 + = 1- quaternion components are related to the Euler angles as follows ... 

The appropriate angles are shown in fig. 1(a), cosi = h. h and the cap indicates a unit vector. The trigonometric functions required in the simulation are all calculated rapidly as vector products. In the case of non-linear molecules Stone [33] expands the potential in an orthonormal set of S-functions. General formulae are given for calculating the forces and torques obtained by differentiating these expansions. 

We will now consider how to define an orientational order parameter. For a cylindrically symmetry molecule, only the angle of orientation is relevant thus the order parameter should be based on a trigonometric function like cosfi. However, an order parameter has to account for the symmetry of the structure it is defined for, and in a nematic phase the molecules are equally likely to be pointing down as they are up i.e. angles and 180° + are equally likely. Since cos(180 + ) = — cos( ), the function should be squared, i.e. cos /3 used, otherwise the average will always be zero. This is a suitable order parameter, but it is not properly normalized, because the... 

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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