Sine and Cosine

Let us focus on one angle of a right triangle, designated by 0 in Fig. 4.7. We designate the two perpendicular sides as being opposite and adjacent to the angle 6. The sine and cosine are then defined as the ratios 

A popular mnemonic for remembering which ratios go with which trigonometric functions is SOHCAHTOA, which might be the name of your make-believe Native American guide through the trigonometric forest. 

Right triangle used to define trigonomefiic functions. 

FIGURE 4.8 Unit circle showing sin 0 and cos 0 in each quadrant. Positive values of sine and cosine extend upward and to the right, respectively. Negative values point downward or to the left 

III and IV, while cos 9 is positive in I and IV and negative in II and III. It should also he clear from the diagram that, for real values of 9, sin0 and cos 9 can have values only in the range [—1,1]. 

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Figure 2 Flow diagram of the DHT with N=8, P=3. Broken lines represent transfer factors -1 while full lines represent unity transfer factor. The crossover boxes perform the sign reversal called for by the shift theorem which also requires the sine and cosine factors Sn, Cn.

Consider a periodic function x(t) that repeats between t = —r/2 and f = +r/2 (i.e. has period t). Even though x t) may not correspond to an analytical expression it can be written as the superposition of simple sine and cosine fimctions or Fourier series, Figure 1.13. 

Now let us eonsider a funetion that is periodie in time with period T. Fourier s theorem states that any periodie funetion ean be expressed in a Fourier series as a linear eombination (infinite series) of Sines and Cosines whose frequeneies are multiples of a... 

Hence we see that this simple periodic function has just two terms in its Fourier series. In terms of the Sine and Cosine expansion, one finds for this same f(t)=Sin3t that an = 0, bn =... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

After the forces are evaluated for each cylinder of a multistage compressor, all forces must be summed in the x and y direction. For the max imum shaking forces, the value of the crank angle, which contributes the maximum force, should be used. This involves taking the respective sine and cosine functions to their maximum. For example, a vertical cylinder will have the maximum component force at a crank angle of 0 and 180 . At this time, the horizontal components, primary and secondary, are zero. 

The use of parity-adapted basis functions in Eq. (30) has several advantages it permits us to use real sine and cosine basis functions for the torsional angle, 4> it allows us to focus only on positive values of K and for the case of = 0, it allows us to divide the calculation into two smaller calculations for each... 

Before discussing the Fourier transform, we will first look in some more detail at the time and frequency domain. As we will see later on, a FT consists of the decomposition of a signal in a series of sines and cosines. We consider first a signal which varies with time according to a sum of two sine functions (Fig. 40.3). Each sine function is characterized by its amplitude A and its period T, which corresponds to the time required to run through one cycle (2ti radials) of the sine function. In this example the frequencies are 1 and 3 Hz. The frequency of a sine function can be expressed in two ways the radial frequency to (radians per second), which is... 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

As explained before, the FT can be calculated by fitting the signal with all allowed sine and cosine functions. This is a laborious operation as this requires the calculation of two parameters (the amplitude of the sine and cosine function) for each considered frequency. For a discrete signal of 1024 data points, this requires the calculation of 1024 parameters by linear regression and the calculation of the inverse of a 1024 by 1024 matrix. 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

The representation of tp(x, t) by the sine function is completely equivalent to the cosine-function representation the only difference is a shift by A/4 in the value of X when t = 0. Moreover, any linear combination of sine and cosine representations is also an equivalent description of the simple harmonic wave. The most general representation of the harmonic wave is the complex function... 

The term Fourier coefficient originates from the theory of Fourier series, in which periodic functions are expanded based on a set of sine- and cosine-functions. The expansion coefficients are called Fourier coefficients. 

Similarly, many different types of functions can be used. Arden discusses, for example, the use of Chebyshev polynomials, which are based on trigonometric functions (sines and cosines). But these polynomials have a major limitation they require the data to be collected at uniform -intervals throughout the range of X, and real data will seldom meet that criterion. Therefore, since they are also by far the simplest to deal with, the most widely used approximating functions are simple polynomials they are also convenient in that they are the direct result of applying Taylor s theorem, since Taylor s theorem produces a description of a polynomial that estimates the function being reproduced. Also, as we shall see, they lead to a procedure that can be applied to data having any distribution of the X-values. 

Without repeating the work, a similar result is obtained for S(x) = cos m. Combination of sine and cosine series leads to the even more general exponential form, such that... 

An important corollary of the principle of superposition is that a wave of any shape can be described mathematically as a sum of a series of simple sine and cosine terms, which is the basis of the mathematical procedure called the Fourier transform (see Section 4.2). Thus the square wave, frequently used in electronic circuits, can be described as the sum of an infinite superposition of sine waves, using the general equation ... 

A widely used example of orthogonal functions is the set of sines and cosines. For example, given any real number a, and the function sin nx for integer values of n, 3) is equal to [a, a+ 2ji]. We can check that... 

Similar orthogonality relationships can be shown for cosines. In addition, sines and cosines are mutually orthogonal, i.e., for any m and n... 

The shorter the wavelength, the faster its decay. Mineral scale heterogeneities in rocks disappear long before meter-scale or even larger heterogeneities. This concept can be extended to any arbitrary combination of periodic functions in Section 2.6, we have already met the idea that any function bounded over an interval can be expanded as a sum of sine and cosine functions. Shorter wavelengths will decay much faster... 

The solution to this equation can be written in terms of sines and cosines ... 

Now the first term on the right is in phase with the applied strain, i.e. it has the form of a sine wave. This can be equated with the storage modulus. Conversely the phase difference between the second term on the right and the applied signal is the difference between sine and cosine waves which can be equated with the loss modulus ... 

Other integrals of importance arc the sine and cosine integrals Ci(a ), Si (a ), which are defined by the equations... 

It is convenient to introduce the sine and cosine transforms P and Q of the impulse response ... 

In practice, the phase shift and the modulation ratio M are measured as a function of co. Curve fitting of the relevant plots (Figure 6.6) is performed using the theoretical expressions of the sine and cosine Fourier transforms of the b-pulse response and Eqs (6.23) and (6.24). In contrast to pulse Jluorometry, no deconvolution is required. 

Data analysis in phase fluorometry requires knowledge of the sine and cosine of the Fourier transforms of the b-pulse response. This of course is not a problem for the most common case of multi-exponential decays (see above), but in some cases the Fourier transforms may not have analytical expressions, and numerical calculations of the relevant integrals are then necessary. 

The procedure in phase-modulation fluorometry is more straightforward. The sine and cosine Fourier transforms of the d-pulse response are, according to Eqs (6.30) and (6.31), given by... 

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