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In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root: = = +
The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc. [1] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc ...
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 ...
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
A ray through the unit hyperbola = in the point (,), where is twice the area between the ray, the hyperbola, and the -axis. The earliest and most widely adopted symbols use the prefix arc-(that is: arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth), by analogy with the inverse circular functions (arcsin, etc.).
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.
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).
There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as sin −1, asin, or, as is used on this page, arcsin. For each inverse trigonometric integration formula below there is a corresponding formula in the list of integrals of inverse hyperbolic functions.