PlanetMath: cyclometric functions

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

cyclometric functions

(Definition)

The trigonometric functions are periodic, and thus get all their values infinitely many times. Therefore their inverse functions, the cyclometric functions, are multivalued, but the values within suitable chosen intervals are unique.

The principal values of the most used cyclometric functions are as follows:

$\arcsin{x}$ is the angle satisfying $\sin y = x$ and $-\frac{\pi}{2} < y \leqq \frac{\pi}{2}$ (defined for $-1 \leqq x \leqq 1$ )
$\arccos{x}$ is the angle satisfying $\cos y = x$ and $0 \leqq y < \pi$ (defined for $-1 \leqq x \leqq 1$ )
$\arctan{x}$ is the angle satisfying $\tan y = x$ and $-\frac{\pi}{2} < y < \frac{\pi}{2}$ (defined in the whole $\mathbb{R}$ )
${\mathrm{arccot}}\,{x}$ is the angle satisfying $\cot y = x$ and $0 < y < \pi$ (defined in the whole $\mathbb{R}$ )

These functions are denoted also $\sin^{-1}x$ , $\cos^{-1}x$ , $\tan^{-1}x$ and $\cot^{-1}x$ . We here use these temporarily for giving the corresponding multivalued functions ( $n = 0,\, \pm1,\, \pm2,\, ...$ ):

$\displaystyle \sin^{-1}x = n\pi+(-1)^n\arcsin{x}$

$\displaystyle \cos^{-1}x = 2n\pi\pm\arccos{x}$

$\displaystyle \tan^{-1}x = n\pi+\arctan{x}$

$\displaystyle \cot^{-1}x = n\pi+{\mathrm{arccot}}\,{x}$

Some formulae:

$\displaystyle \arcsin{x}+\arccos{x} = \frac{\pi}{2}$

$\displaystyle \arctan{x}+{\mathrm{arccot}}\,{x} = \frac{\pi}{2}$

$\displaystyle \arcsin{x} = \int_0^x\frac{dt}{\sqrt{1-t^2}}\,dt$

$\displaystyle \arctan{x} = \int_0^x\frac{dt}{1+t^2}\,dt$

$\displaystyle \arcsin{x} = x+\frac{1}{2}\!\cdot\!\frac{x^3}{3}+ \frac{1\!\cdot\... ...!5}{2\!\cdot\!4\!\cdot\! 6}\!\cdot\!\frac{x^7}{7}+...\quad(\vert x\vert\leqq 1)$

$\displaystyle \arctan{x} = 1-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+-... \quad(\vert x\vert \leqq 1)$

$\displaystyle \frac{d}{dx}\arccos{x} = -\frac{1}{\sqrt{1-x^2}}\quad(\vert x\vert < 1)$

$\displaystyle \frac{d}{dx}{\mathrm{arccot}}\,{x} = -\frac{1}{1+x^2}\quad(\forall x\in \mathbb{R})$

The classic abbreviations of the cyclometric functions are usually explained as follows. The values of the trigonometric functions are got from the unit circle by means of its arc (in Latin arcus) with starting point (1,0). The arc represents the angle (which may be thought as a central angle of the circle), and its end point $(\xi,\,\eta)$ is achieved when the starting point has circulated along the circumference anticlockwise for positive angle (and clockwise for negative angle). Then the cosine of the arc (i.e. angle) is the abscissa $\xi$ of the end point, the sine of the arc is the ordinate $\eta$ of it. Correspondingly, one can get the tangent and cotangent of the arc by using certain points on the tangent lines and of the unit circle.

The word cosine is in Latin cosinus, its genitive form is cosini. So e.g. the $\arccos$ of a given real number means the `arc of the cosine value ', in Latin arcus cosini x.