Trigonometric identities

The relations in Eq. C.4 may be added or subtracted to generate other useful expressions  

A vector P is a quantity with both magnitude and direction, usually designated in boldface type (e.g., B, to), and represented pictorially as an arrow /. A vector in a space of N dimensions may be expressed in terms of N orthogonal unit basis vectors as a linear ordered array of numbers that describe the contributions from 

The scalar (dot) product of two vectors P and R at an angle 0 to each other is given by 

The vector (cross) product of vectors P and R is a vector Q orthogonal to both P and R of magnitude given by 

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Tsai and Pagano [2-7] ingeniously recast the stiffness transformation equations to enable ready understanding of the consequences of rotating a lamina in a laminate. By use of various trigonometric identities between sin and cos to powers and sin and cos of multiples of the angle, the transformed reduced stiffnesses. Equation (2.85), can be written as... 

Using trigonometric identities, the off-diagonal elements may be written as... 

The solution becomes particularly simple and time-invariant if r is chosen in such a way that the cross-terms vanish. Using basic trigonometric identities, we cancel the off-diagonal term... 

Expanding this result and applying trigonometric identities, one obtains... 

Other concepts we will use freely include parhal derivatives, trigonometric identities, the natural numbers N = 1, 2, 3,. .., basic properties of integra-hon and proof by induction. Interested readers will hnd a nice introduction to proof by induction in [Sp, Chapter 2]. 

For = 2 we must make use of some trigonometric identities to see that... 

The relatively simple inversion of the direction-cosine matrix can be seen from trigonometric identities among direction cosines, stated as... 

Thus the polar normal coordinate

linear molecule. Substitution of (6.89) into (6.88) and use of trigonometric identities gives... 

Be familiar with the use of trigonometric identities and addition formulae... 

Working with the properties of logarithms and trigonometric identities. 

Finally, applying the appropriate trigonometric identity, we obtain... 

Equation 7.A2.6 can be put into a more useful form by using the trigonometric identity ... 

The nonlinear polarizabilities in the classical spring problem arise from anharmonic contributions to the spring constant. Resolution of eq. 3 into harmonics of frequency nu using trigonometric identities provides an understanding of how specific orders of anharmonicity in V(x) lead to anharmonic polarizations at frequencies different from that of the applied field S(t). In the classical problem, the coefficients an are determined by the anharmonicity constants in V(x) [10]. 

These relations are expressed through the following trigonometric identity... 

One way to achieve a harmonic series is to set 0 = b. Then the trigonometric identity becomes... 

Here, a0 is the polarizability at the equilibrium position, and (daJdq)0 is the change rate of a with respect to the change in q, evaluated at the equilibrium position. Combining Eqs. (16.13)-(16.15) and the trigonometric identity,... 

Understanding how the chemical shift and /-coupling modulation in t works (s s for the crosspeak and s c for the diagonal peak) takes a bit of mathematical manipulation. The sin( 2ati) sin( r/fi) term (s s ) can be written as a sum rather than a product using the trigonometric identity cos(a+fi) = cosa cos/J—sina sin/i ... 

By combining the exponentials and employing the trigonometric identity 1 - c.osx = 2sinz(x/2), one obtains for the amplification factor ... 

Using the trigonometric identity cos2x = (l/2)(l+cos(2x)), one sees the argument for second harmonic detection ... 

Calculate the first-order perturbation correction to the ground-state energy level using the particle in a box with V(x) = 0 for 0 x a, V(x) = oo for x < 0, x > a as the unperturbed system. Then calculate the first-order perturbation correction to the ground-state wave function, terminating the expansion after the term k = 5. (See Appendix A for trigonometric identities and integrals.)... 

We use the fact that IZIZ = 1, ignoring the normalizing factor.) Of the final eight terms, four represent antiphase magnetization, which is not observed, and can thus be deleted. After applying well-known trigonometric identities, we obtain... 

We can picture the peaks more easily if we use trigonometric identities to obtain for expressions (3) and (4), that is, one diagonal and one cross peak, the following ... 

Verify the derivation of Eq. 12.6 and 12.7 by using trigonometric identities from Appendix C. 

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