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The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that ...
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same.
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.
A second of arc, arcsecond (arcsec), or arc second, denoted by the symbol ″, [2] is 1 / 60 of an arcminute, 1 / 3600 of a degree, [1] 1 / 1 296 000 of a turn, and π / 648 000 (about 1 / 206 264.8 ) of a radian.
Following this recommendation, the ISO 80000-2 standard abbreviations use the prefix ar-(that is: arsinh, arcosh, artanh, arsech, arcsch, arcoth). In computer programming languages, inverse circular and hyperbolic functions are often named with the shorter prefix a-(asinh, etc.). This article will consistently adopt the prefix ar-for convenience.
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
arcsec – inverse secant function. arcsin – inverse sine function. arctan – inverse tangent function. arctan2 – inverse tangent function with two arguments. (Also written as atan2.) arg – argument of. [2] arg max – argument of the maximum. arg min – argument of the minimum. arsech – inverse hyperbolic secant function.
The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem.