Arcsine function

Over what range and domain can the arcsine function be defined if we want it to be monotonically increasing ... 

This can be read as x is the angle whose sine is y. The arcsine function is also called the inverse sine function, and another notation is also common ... 

Sketch graphs of the arcsine function, the arccosine function, and the arctangent function. Include only the principal values. g]... 

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. 

Here we have a case where all, not only most, of the area under the probability density function is located within V 2 standard deviations of the mean, but where this fact alone gives a very misleading picture of the arcsine distribution, whose area is mainly concentrated at the edges of the distribution. Quantitatively, this is borne out by the easily verified fact that one half of the area is located outside of the interval [—0.9,0.9]. 

Arcsine distribution, 105, 111 Assumption of molecular chaos, 17 Asymptotic theory, 384 of relaxation oscillations, 388 Asynchronous excitation, 373 Asynchronous quenching, 373 Autocorrelation function, 146,174 Autocovariance function, 174 Autonomous problems, 340 nonresonance oscillations, 350 resonance oscillations, 350 Autonomous systems, 356 problems of, 323 Autoperiodic oscillation, 372 Averages, 100... 

As shown in Sec. 6.4 radiation is totally reflected at the boundary between a medium with a refractive index n and a medium with lower refractive index ni if it hits this boundary with an incident angle greater than the critical angle 6>,. = arcsin(/72//J ). The reflected radiation energy penetrates the boundary as a so-called evanescent wave. The penetration depth dp is the thickness within which the intensity decreases to 1/e of the intensity at the boundary (see Fig. 6.5-2). It is a function of the refractive indices n and H2, the incident angle 0, and the wavelength A (in vacuum). With ri2 = ri2/n ... 

Some forms of BASIC may not have an arccosine (ACOS) function, even though they all have an arctangent function, ATAN or ATN. Perhaps this omission has come about because arccosine (and arcsine) can be expressed in terms of arctan. In fact, some BASIC manuals give the explicit formulas for computing arccosine and arcsine. 

The inverse functions of the trigonometric functions (on suitably chosen domains) are denoted as arccos x, arcsin x, arctan x, and arccot x. Important relations between the trigonometric functions and/or their inverse functions are... 

Built-in functions and operators sqrt () - square root, abs () - absolute value, - factorial, sin() - sine, cos() - cosine, tan() - tangent, asin() - arcsine, acos () - arccosine h atan () - arctangent. It is possible to work with other functions if they have been defined before. 

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