# cyclometric functions

The of the most used cyclometric functions are as follows:

• $\arcsin{x}$  is the angle $y$ satisfying  $\sin y=x$  and  $-\frac{\pi}{2}  (defined for $-1\leqq x\leqq 1$)

• $\arccos{x}$  is the angle $y$ satisfying  $\cos y=x$  and  $0\leqq y<\pi$  (defined for $-1\leqq x\leqq 1$)

• $\arctan{x}$  is the angle $y$ satisfying  $\tan y=x$  and  $-\frac{\pi}{2}  (defined in the whole $\mathbb{R}$)

• $\operatorname{arccot}\,{x}$  is the angle $y$ satisfying  $\cot y=x$  and  $0  (defined in the whole $\mathbb{R}$)

Those functions  are denoted also $\sin^{-1}x$, $\cos^{-1}x$, $\tan^{-1}x$ and $\cot^{-1}x$.  We here use these notations temporarily for giving the corresponding multivalued functions ($n=0,\,\pm 1,\,\pm 2,\,...$):

 $\sin^{-1}x=n\pi+(-1)^{n}\arcsin{x}$
 $\cos^{-1}x=2n\pi\pm\arccos{x}$
 $\tan^{-1}x=n\pi+\arctan{x}$
 $\cot^{-1}x=n\pi+\operatorname{arccot}\,{x}$

Some formulae

 $\arcsin{x}+\arccos{x}=\frac{\pi}{2}$
 $\arctan{x}+\operatorname{arccot}\,{x}=\frac{\pi}{2}$
 $\arcsin{x}=\int_{0}^{x}\frac{dt}{\sqrt{1-t^{2}}}\,dt$
 $\arctan{x}=\int_{0}^{x}\frac{dt}{1+t^{2}}\,dt$
 $\arcsin{x}=x+\frac{1}{2}\!\cdot\!\frac{x^{3}}{3}+\frac{1\!\cdot\!3}{2\!\cdot\!% 4}\!\cdot\!\frac{x^{5}}{5}+\frac{1\!\cdot\!3\!\cdot\!5}{2\!\cdot\!4\!\cdot\!6}% \!\cdot\!\frac{x^{7}}{7}+\ldots\quad(|x|\leqq 1)$
 $\arctan{x}=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+-\ldots\quad(|x|% \leqq 1)$
 $\frac{d}{dx}\arccos{x}=-\frac{1}{\sqrt{1-x^{2}}}\quad(|x|<1)$
 $\frac{d}{dx}\operatorname{arccot}\,{x}=-\frac{1}{1+x^{2}}\quad(\forall x\in% \mathbb{R})$

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 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$x=1$  and  $y=1$  of the unit circle.

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

 Title cyclometric functions Canonical name CyclometricFunctions Date of creation 2013-03-22 14:36:00 Last modified on 2013-03-22 14:36:00 Owner pahio (2872) Last modified by pahio (2872) Numerical id 34 Author pahio (2872) Entry type Definition Classification msc 26A09 Synonym arc functions Synonym arcus functions Synonym inverse trigonometric functions  Related topic Trigonometry Related topic ComplexSineAndCosine Related topic TaylorSeriesOfArcusSine Related topic TaylorSeriesOfArcusTangent Related topic AreaFunctions Related topic RamanujansFormulaForPi Related topic SawBladeFunction Related topic TerminalRay Related topic DerivativeOfInverseFunction Related topic LaplaceTransformOfFracftt Related topic OstensiblyDiscontinuousAntiderivative Related topic I Defines branch Defines principal branch Defines sine Defines cosine Defines arc sine Defines arc cosine Defines arc tangent Defines arc cotangent Defines inverse sine Defines inverse tangent