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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 ...
In the integral , we may use = , = , = . Then, = = () = = = + = +. The above step requires that > and > We can choose to be the principal root of , and impose the restriction / < < / by using the inverse sine function.
When the direction of a Euclidean vector is represented by an angle , this is the angle determined by the free vector (starting at the origin) and the positive -unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive x ...
In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: [1] = + + = = + +. This series converges in the complex disk | |, except for = (where =).
The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions.For a complete list of integral formulas, see lists of integrals.
The tangent half-angle substitution relates an angle to the slope of a line. Introducing a new variable = , sines and cosines can be expressed as rational functions of , and can be expressed as the product of and a rational function of , as follows: = +, = +, = +.
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The inverse tangent integral is a special function, defined by: Ti 2 ( x ) = ∫ 0 x arctan t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt} Equivalently, it can be defined by a power series , or in terms of the dilogarithm , a closely related special function.