Functions trigonometric

The ordinary trigonometric functions include the sine, the cosine, the tangent, the cotangent, the secant, and the cosecant. These are sometimes called the circular trigonometric functions to distinguish them from the hyperbolic trigonometric functions discussed briefly in the next section of this chapter. 

The trigonometric functions of the angle 2 are defined in the same way, except that as drawn in Fig. 2.1, the distance X2 must be counted as negative, because the point B2 is to the left of A. If the point B2 were below A, then y2 would also be counted as negative. 

This does not look like a correct equation until we understand that on the left-hand side the angle is measured in degrees and on the right-hand side the angle is measured in radians. If you use degrees, you should always include the degree sign 

The trigonometric functions are examples of mathematical functions. A mathematical function is a rule that provides a unique coimection between the value of one variable, called the independent variable or the argument of the function, and another variable, which we call the dependent variable. When we choose a value for the independent variable, the function provides a corresponding value for the dependent variable. For example, if we write 

Equations (2.10) and (2.12) express the fact that the sine and the tangent are odd functions, and Eq. (2.11) e5q)resses the feet that the cosine is an even function. If /(x) is an odd function, then 

We can give the same label 9 to the angle that creates this arc. In this case, we refer to the angle in units of radians, and thus 2tt corresponds to a complete circle. Radians might seem superficially to be an inconvenient unit for measuring angles. In fact, they turn out to be the most natural unit, as we will see when we discuss derivatives in the next chapter. 

FIGURE 1.1 Definition of the sine and cosine function, in terms of positions on a circle with radius 1. 

Don t let the definition based on tracking around a circle fool you—sine and cosine waves appear in many problems in chemistry and physics. The motion of a mass 

FIGURE 1.2 Sine (y) and cosine (x) components of motion at a constant angular velocity co along a circular path. 

momentum, velocity and acceleration are examples of vector quantities (they have a direction and a magnitude) and are written in this book with an arrow over them. Other physical quantities (for example, mass and energy) which do not have a direction will be written without an arrow. The directional nature of vector quantities is often quite important. Two cars moving with the same velocity will never collide, but two cars with the same speed (going in different directions) certainly might  

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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 ... 

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... 

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... 

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. 

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. 

For definitions of trigonometric functions, see Trigonometry. ) Right triangle (Figure 1-1)... 

Legitimate operations on equations include addition of any quantity to both sides, multiplication by any quantity of both sides (unless this would result in division by zero), raising both sides to any positive power (if is used for even roots) and taking the logarithm or the trigonometric functions of both sides. 

The basic trigonometric functions are the sine, cosine, and tangent. 

Trigonometric Function Values at Major Angle Values... 

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

Thus the original differential equation (6-90) of the second order has been replaced by the system (6-96) of two first order differential equations in terms of the amplitude a and the phase 9. Moreover, as Eqs. (6-96) contain the small factor (i on the right-hand side, the quantities, a and 9 are small, that is, both a and 9 are slowly varying functions of time and one can assume that during one period T = 2nfca, the trigonometric functions vary but slightly. 

Correspondingly, the parameter 1/8q tends to zero and we can expand the inverse trigonometric function tan 1/8q in the power series... 

The solutions to equation (2.34) are functions that are proportional to their second derivatives, namely sin(27rv/A) and cos(2jrjc/A). The functions exp[27riv/A] and exp[—27riv/A], which as equation (A.31) shows are equivalent to the trigonometric functions, are also solutions, but are more difficult to use for this system. Thus, the general solution to equation (2.34) is... 

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