Cosine formula

Use the triangle rule in the form c = a + A to derive the form of the cosine formula involving the angle C. 

The summation for the coefficient of the P2 term likewise includes phase insensitive contributions where I m = Im, but also now one has terms for which I = I 2, which introduce a partial dependence on relative phase shifts— specifically on the cosine of the relative phase shift. Again, this conclusion has long been recognized for example, by an explicit factor cos(ri j — ri i) in one term of the Cooper-Zare formula for the photoelectron (3 parameter in a central potential model [43]. 

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 ... 

Expressions for the sine and cosine of the sum or difference of two angles are given by the following formulae ... 

When used as the dispersion formula for the phonons and polaritons in orthorhombic crystals, the symbols in Eq. (11.22) have the following meaning r)= 1,2,3 designates the three directions of the principal orthogonal axes. sv are the direction cosines of the normalized wave vector s = k/k with respect to the three principal axes of the crystal. If the unit vectors in the directions of these three principal axes are designated eue2,e3, one can write... 

Three roots of the cubic equation are coefficients Bn, B22 and B33. The cubic equation is solved by applying Kardan s formula. When we know Bn, B22 and B33 values, we can determine values of cosine angle of axes rotation, and thereby the system of equations that connects old and new coordinates. 

Two very useful formulas for converting between sines and cosines of different angles are ... 

The hindering in rotation necessitates the introduction of an extra correction factor in the formula for (ro2). This factor contains the quantity (cos rotation angles. For free rotation (cos tp) = 0 (all values for (p are equally probable), but it differs from 0 when potential barriers are present. It can be calculated from the potential curves. The formula becomes now... 

The Slater and Koster (1954) tables of interatomic matrix elements as functions of the direction cosines, /, m, and /i, of the vector from the left state to the right state. Other matrix elements are found by permuting indices. General formulae for these expressions and explicit expressions involving/ and r/ orbitals have been given recently by Sharma (1979). 

Let a be the angle (less than two right angles) that the direction of the force F makes with the direction MM of the displacement of the material point by the definition of the cosine of an angle, the work we have just described will be represented in magnitude and direction by the formula... 

A contour plot is shown in Fig. 7.8. Note that this function is cylindrically-symmetrical about the z-axis with a node in the x, y-plane. The eigenfunctions 21 1 are complex and not as easy to represent graphically. Their angular dependence is that of the spherical harmonics 7i i, shown in Fig. 6.4. As deduced in Section 4.2, any linear combination of degenerate eigenfunctions is an equally-valid alternative eigenfunction. Making use of the Euler formulas for sine and cosine,... 

When the angles are all 90° (so that their cosines are 0), this formula reduces to the simple result V = abc for the volume of a rectangular box. If the mass of the unit cell contents is known, the theoretical cell density can be computed. This density must come close to the measured density of the crystal, a quantity that can be... 

It has been assumed that the movement correlates to a sine function and the velocity to a cosine function. As this would lead to considerable integration errors the instrument referred to has been provided with a cosine corrector this corrects the measuring signal coming from the photomultiplier after it has been amplified. According to Ebel et al.17), however, the velocity of the oscillating movement corresponds to an entirely different function (formula and figure can be found in the... 

The cosine of the sun s zenith angle 5 depends on time, latitude, and longitude. The optical path length t exhibits a seasonal cycle mainly due to a variable content of water vapor. It should be noted that this formula refers to the present state of the atmosphere and may need modification to simulate the Baltic Sea of the past or future. 

Another approximate formula may also be derived for the cosine potential as follows. On expansion of and pe in Taylor s series, we have from Eqs. (9) and (22)... 

This intersection corresponds to a fixed point, since x = cos% and therefore /(. ) = 0. Moreover, when the line lies above the cosine curve, we have x >cosx and so x > 0 the flow is to the right. Similarly, the flow is to the left where the line is below the cosine curve. Hence x is the only fixed point, and it is unstable. Note that we can classify the stability of x, even though we don t have a formula for x itself ... 

We are now able to take into account the perturbation effect of more Ugands upon our standard basis, the unprimed basis. This is done by introducing assumption III which states that we have only to sum the expressions of Eq. (7) over k. It is important to observe that each new Ugand corresponds to a new position of the Z axis and thereby to a new primed basis set, but this basis set has been eliminated from the formulae, which only contain the direction cosines ( , yic) of each Z axis and thereby are fuUy characterized by the astronomical (directional) coordinates of each hgand. The final formula for equal hgands is... 

The change from cosine to sine modulation in the EXSY experiment can be though of as a phase shift of the signal in tl. Mathematically, such a phase shifted cosine wave is written as cos(f2jtj +(/)), where (f) is the phase shift in radians. This expression can be expanded using the well known formula cos( + B) = cos A cos B - sin A sin B to give... 

Let us now examine the formulas of transformation necessary to deal with cases where the coordinate system for the equations of flow is not identical with the coordinate system for the stress components. At the point P let x, g and z be in the directions of the principal axes of distortion as defined by Eqn 3-7 and let a, fa, and c be the partial derivatives of flow along these axes. Let x, g and z be a set of orthogonal axes whose orientation is given by the matrix of direction cosines shown below ... 

As in the case of Huckel-type tt systems, it is possible to give a convenient mnemonic form for the closed formulas 118> which is based on the relation of cosine and sine functions to motion on a circle. 

EXAMPLE 4.7 Find the derivative of tan(ax) by using the formulas for the derivatives of the sine and cosine. 

Where A, A, v, t, and 0are the amplitude, wavelength, velocity, time, and phase, respectively. This equation describes the propagation of a cosine curve [A cos(2jt/A)x] along the x-axis. Introducing the frequency v = v/A, the general formula of wave can be written as ... 

These series are employed for calculating the numerical values of angles between 0 and tt. All the other angles found in trigonometrical tables of sines and cosines, can be then determined by means of the formulas, page 611,... 

Obtain cosine values, by interpolation, for 22° using Lagrange s three-point interpolation formula. Evaluate the error of your result. Compare the error with its correct value. 

Here we will introduce three similarity coefficients that have been widely used for both realvalued (i.e. continuous) and binary (dichotomous) descriptors the Tanimoto coefficient, the Dice coefficient and the Cosine coefficient The formulae used to compute these coefficients are given in Table 12.3, where, for completeness, we have also provided the Euclidean and Hamming expressions that were introduced in Section 9.13. Different expressions are used for real-valued data (where the molecule is represented by a vector containing N real values Xj) and for binary data (where each molecule is represented by N binary values). For binary data, we additionally define a to be the number of bits on in the bitstring for A, b to be the number of bits on in the bitstring for B, and c to be the number of bits that are on in both A and B (calculated using the AND operator). 

The two kinds of Bessel functions, thus, have the asymptotic dependence of slowly damped cosines and sines. In analogy with Euler s formula = cos X i sinx, we define Hankel functions of the first and second kinds ... 

---