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Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios.
The inverse trigonometric functions are also known as the "arc functions". C is used for the arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives. There are three common notations for inverse trigonometric ...
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
Generally, if the function is any trigonometric function, and is its derivative, ∫ a cos n x d x = a n sin n x + C {\displaystyle \int a\cos nx\,dx={\frac {a}{n}}\sin nx+C} In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration .
Pages in category "Inverse trigonometric functions" The following 47 pages are in this category, out of 47 total. This list may not reflect recent changes. ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus). [9] [10] For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin(x). [9] [10] Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ārea). [10]
The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, . [12] By comparison, the versed sine of a curved track ...