Trigonometric Notation

Starting with a sinusoidal input of strain in a Maxwell element (Section 9.2), we derive the resulting sinusoidal stress. First, we let the strain s be a function of a maximum strain e and time t with a frequency co  

This is a simple linear differential equation of the form  

When P and Q are functions of x only, the general solution of such an equation is 

Under steady-state conditions for which t/X 1, the second term on the right-hand side of the above equation drops out leaving 

Substituting the above equation into Eiquation 9.A. 10, we arrive at 

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Since not all electronic calculators are alike, detailed instructions cannot be given here. Read your instruction manual. You should purchase a calculator which, in addition to , —, x, and a functions, provides at least the following scientific notation (powers of ten) logarithms and antilogarithins (inverse logarithms) both natural and common (base ten) and exponentials (y ). If it has these functions, it will probably have reciprocals (1/jt), squares, square roots, and trigonometric functions as well. 

The calculated value for the variance of our new variable A3 confirms that there is an increased spread of the data on the new axis. As well as this algegbraic notation, it is worth pointing out that the coefficients of the normalized linear combination may be represented by the trigonometric identities... 

Notice the common notation for a power of a trigonometric function the exponent is written after the symbol for the trigonometric function and before the parentheses enclosing the argument. 

Now since the product of powers of x etc. is merely compact notation for a product of powers of trigonometric functions, we can expand the real spherical harmonics in terms of i, y, z to generate a sum of integrals which are entirely products of powers of these angular variables, and use the fact that... 

To compound the notational confusion, sin x and cos x are used to designate inverse trigonometric functions, also written as arcsinx and arccosx, respectively. These inverse functions are related by the following correspondences ... 

This notation simplifies considerably the mathematical expressions, and there is no need to use complicated trigonometric formulae when dealing with the changes in the phase and ampUtude of AC signals. 

Every chemistry student uses a calculator to solve chemistry problems. A suitable calculator can (1) add, subtract, multiply, and divide (2) perform these operations in exponential notation (3) work with logarithms and (4) raise any base to a power. Calculators that can perform these operations usually have other capabilities, too, such as finding squares and square roots, carrying out trigonometric functions, and offering shortcuts for pi and percentage, enclosures, statistical features, and different levels of storage and recall. 

Notice the common notation for a power of a trigonometric function the exponent is written after the symbol for the trigonometric function and before the parentheses enclosing the argument. Do not use this notation if the exponent is —1, since this notation is used for the inverse trigonometric functions, which we discuss later. 

The — 1 superscript indicates an inverse function. It is not an exponent, even though exponents are written in the same position. If you need to write the reciprocal of sin(y), you should write [sin (y)] to avoid confusion. It is probably better to use the notation of Eq. (2.42) rather than that of Eq. (2.43). The other inverse trigonometric functions such as the inverse cosine and inverse tangent are defined in the same way as the arcsine function. 

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