Arcsinus

De arcsinus, ook boogsinus, aangeduid door $\mathrm {asin} ,\arcsin ,\mathrm {bgsin}$ ^[1] of $\sin ^{-1}$ ,^[2] is een cyclometrische functie in de wiskunde die de inverse functie is van de sinus indien het domein daarvan beperkt wordt tot het interval $[-\pi /2,\pi /2]$ . Deze beperking is nodig vanwege het periodieke karakter van de sinus. Het resultaat van de arcsinus is de hoek tussen $-\pi /2$ en $\pi /2$ waarvan de sinus het argument als waarde heeft. Het domein is $[-1,1]$ en het bereik is $[-\pi /2,\pi /2]$ .

De grafiek van $y=\arcsin x$ is het spiegelbeeld van de grafiek van de beperkte sinus ten opzichte van de lijn $y=x$ .

Definitie

De functie $\arcsin$ is gedefinieerd voor $x\in [-1,1]$ door de relatie

\arcsin(x)=\alpha \quad \Longleftrightarrow \quad \alpha \in [-{\tfrac {1}{2}}\pi ,{\tfrac {1}{2}}\pi ]{\mbox{ en }}\sin(\alpha )=x

In woorden: de hoek of boog waarvan de sinus $x$ is, is gelijk aan $\alpha$ .

Machtreeks

De arcsinus heeft de reeksontwikkeling:

\arcsin(x)=\sum _{n=0}^{\infty }{\frac {\Gamma (n+{\frac {1}{2}})}{{\sqrt {\pi }}(2n+1)n!}}x^{2n}

Daarin is $\Gamma$ de gammafunctie.

Afgeleide

De afgeleide van de arcsinus is:

{{\rm {d}} \over {\rm {d}}x}\arcsin(x)={1 \over {\sqrt {1-x^{2}}}}

voor $x\in (-1,1)$ .

Voetnoten

↑ Alleen in Nederlandstalige teksten en daarom afgeraden
↑ Deze notatie kan verwarring veroorzaken met 1/sin.

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