Trigonometric relations

This integral can be evaluated with the aid of the appropriate trigonometric relations. Furthermore, the upper limit can be replaced by f, as t < t. The result,... 

The coordinates corresponding to the coefficients B m satisfy our requirements. The geometrical meaning of this transformation becomes apparent, when we substitute (18) into expression (15) for o. Denoting by i , the polar angle and azimuth referred to the direction x and the Xy y plane, we obtain, with the aid of well-known trigonometrical relations. 

Even more interesting sounds can be made by more complex usage of the FM formulas. With frequency modulation one might select more than one modulating waveform, or perhaps different waveforms than sinusoids. In addition, a complex amplitude modulation can be imposed. For example, one possibility is revealed in the trigonometric relation... 

It often happens that the coordinates are not transformed simply into each other by a symmetry operation. Trigonometric relations must be used to express, for instance, the consequences of three-fold rotation. 

With the help of standard trigonometric relations, aU) may be rewritten as... 

A root of the quartic equation (2.208) is given in this case by the simple trigonometric relation ... 

It can be further shown (Section 2.9.2) that the trigonometric relation (Equation 2.214) is equivalent to Coulson s hybridization condition (Coulson, 1961 Magnasco, 2007) provided the lone pair hybrid / = hys is Schmidt-orthogonalized against the bond hybrids b = hy and bi = hj2 directed towards the hydrogen atoms Hi and H2. 

But the geometry of Figure 15.7 shows that the blade velocity may also be expressed by the trigonometrical relations ... 

The last formula thus resembles the well-known trigonometrical relation cos2 + sin2 = 1. Draw PM a tangent to the circle AP at P. Drop a perpendicular PM on to the -axis. Let the angle MOP = 0. 

The elliptic functions bear important analogies with the ordinary trigonometrical functions. The latter may be regarded as special forms of the elliptic functions with a zero modulus, and there is a system of formulae connecting the elliptic functions to each other. Many of these bear a formal resemblance to the ordinary trigonometrical relations. Thus,... 

Trigonometric relations between unit vectors in rectangular and spherical coordinates yield the following expression for 8 (see Bird et al 2002, p. 828). Hence,... 

As an example, with the backup of appropriate software and the application of trigonometric relations, relative knee angles for 200 samples of gait measurement for 8 s are demonstrated in Figure 25.11. 

It is instructive to write the stress in an alternative form displaying the amplitude o-°(co) of the stress and the phase angle 5(co) between stress and strain. From trigonometric relations,... 

Rearrangement of (10.29) yields, after applying some trigonometric relations, 2 4... 

Taking into account some trigonometric relation we arrive at... 

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