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That is, if is an arcsine-distributed random variable, then (,). By extension, the arcsine distribution is a special case of the Pearson type I distribution . The arcsine distribution appears in the Lévy arcsine law , in the Erdős arcsine law , and as the Jeffreys prior for the probability of success of a Bernoulli trial .
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
[1] [10] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 (x), Cos −1 (x), Tan −1 (x), etc. [11] Although it is intended to avoid confusion with the reciprocal, which should be represented by sin −1 (x), cos −1 (x), etc., or, better, by ...
Alternatively, notice that whenever θ has a value such that l sin θ ≤ t, that is, in the range 0 ≤ θ ≤ arcsin t / l , the probability of crossing is the same as in the short needle case. However if l sin θ > t, that is, arcsin t / l < θ ≤ π / 2 the probability is constant and is equal to 1.
The opposite leg, O, is approximately equal to the length of the blue arc, s. Gathering facts from geometry, s = Aθ , from trigonometry, sin θ = O / H and tan θ = O / A , and from the picture, O ≈ s and H ≈ A leads to: sin θ = O H ≈ O A = tan θ = O A ≈ s A = A θ A = θ . {\displaystyle \sin \theta ={\frac ...
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
The third arcsine law states that the time at which a Wiener process achieves its maximum is arcsine distributed. The statement of the law relies on the fact that the Wiener process has an almost surely unique maxima, [1] and so we can define the random variable M which is the time at which the maxima is achieved. i.e. the unique M such that
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n).