Law of cosines

A cross-linked polymer has a density of 0.94 g cm" at 25°C and a molecular weight between crosslinks of 28,000. The conformation of one bond in the middle of the molecule changes from trans to gauche, and the molecule opens up by 120°. In w-butane, the trans to gauche transformation requires about 3.3 kJ mol". Estimate a value for AH of stretching based on this model, and use the law of cosines to estimate the magnitude of the opening up that results. 

We can use the law of cosines to estimate the effect on the end-to-end distance as follows. Picture two vectors of equal length b extending from the center of the chain to each of the ends. In the trans conformation of the center bond the... 

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

Solution Use the law of cosines to determine the length of the two diagonals of the substrate unit mesh ... 

The quantity (cosQi,i+j) can be evaluated with the aid of the law of cosines from spherical trigonometry ... 

FIGURE 6.1 Coordinates for the HJ molecular ion. The two nuclei are located along the z-axis, separated by the distance / ab- The coordinates ca and are the distances of the electron from nuclei A and B, respectively Ca and range from 0 to < . The angle 6 is determined from Ca, Cb, and / ab by the law of cosines it does not appear explicitly in the calculation. The angle 4> varies from 0 to 27t. 

Fig. 3.2.2. Light of polarization n4 and wave vector k is scattered in all directions. Only scattered light of wave vector k/ and polarization nf arrives at the detector. The scattering vector q = kt—kf is defined by the geometry. Since the scattered wave has essentially the same wavelength as the incident wave, kf (2n n)lki = ki, it follows from the law of cosines that q = 2ki sin 9/2.

SO the delay lines cannot compensate with a simple linear increment as in Figure 25.10. It can be shown easily that the differential time delay between channels can be determined from the law of cosines. 

If the bond lengths CA and CS are known and the interligand distance AB can be predicted from the 1,3-radii listed in Table 7.2, then the bond angle / can be calculated using Equation (7.1). [Reproduced from http //en.wikipedia.org /wiki/Law of cosines (accessed December 15, 2013).]... 

Here, the left side is fixed under linear isometry, and the squares x x = x and y y = lyp on the right hand side are fixed as well, so that x y is also retained. Moreover, the law of cosines gives the identity... 

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