Plane trigonometric function

The trigonometric functions of angles are the ratios between the various sides of the reference triangles shown in Fig. 3-38 for the various quadrants. Clearly r = Vx2 + y2 > 0. The fundamental functions (see Figs. 3-39, 3-40, 3-41) are Plane Trigonometry... 

Since the exponential function may be defined everywhere in the complex plane, we may expand exp(i0) and, using the series expansions for the trigonometric functions, obtain Euler s formula... 

Another important class of functions encountered in chemistry and physics is the trigonometric functions. Consider the equation x2 + y2 = 1. The set of all points in a plane that satisfy this equation is a circle with radius 1 (Figure 1.1). Any position on the circle could be labeled by the length 9 of the arc which stretches counterclockwise from the positive x-axis to that point. Since the circle has circumference 2jt, only values of 6 between 0 and 2tt are needed to describe the whole circle. 

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. 

Triangles can be constructed either on planes or on spheres. In addition, the six trigonometric functions are used apart from any direct reference to triangles. 

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

[Pg.1354]

There are many practical applications in engineering for the trigonometric functions sine, cosine, tangent, secant, cosecant, and cotangent. These functions are defined as the ratios of the sides of plane right triangles. These functions are shown in Table 7.12. 

These functions are called circular or trigonometric functions. Note that Equations (2-25) are just the transformation Equations (1-4) with r = 1. It is interesting to compare the graphs of functions, such as sin 0 and cos 0, in linear coordinates (coordinates in which 0 is plotted along one axis) to those in plane polar coordinates. Consider, for example, the equation r = A cos 0y where A is a constant. Such an equation can be used to describe the wave properties of p-type atomic orbitals in two dimensions. The functional dependence of r upon 0 can be seen in Table 2-1. 

In the case of particle-plane or two coplanar particle configurations, one has (p = 0 and Eq. (43) simplifies to the form derived by Adamezyk et al. [32]. Despite its apparent simplicity, it is very inconvenient to apply Eq. (43) for three-dimensional situations because of mathematical difficulties in finding the points of the minimum separation of the two bodies involved as a function of their mutual orientation and, consequently, to determine h . Even for such simple particle shapes as spheroids, one has to solve a high-order nonlinear trigonometric equations, which can only be done in an efficient way by iterative methods [15]. 

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