Cosécante
csc α
=
hypoténuse
jambe opposée
=
OB
AB
=
OB"
OA"
=
OB"
OB
=
R
yB
Si hypoténuse (OB, R) = 1,
csc α = OB"
Fonction
`y = text(csc) x`
Domaine
`D(text(csc) x) = R - {pin, (n in Z)}`
Plage de valeurs
`E(text(csc) x) = [-infty, -1] uu [1, infty]`
Parité
`étrange`
Périodicité
`T = 2pi`
Zéros
`non`
Intervalles de constance
`text(csc) x < 0,`
`x in (-pi+2pin, 2pin), n in Z`
`text(csc) x > 0,`
`x in (2pin, pi + 2pin), n in Z`
Intervalles de monotonie
`uarruarruarr x in (-pi+2pin, -pi/2 + 2pin]uu`
`uu[pi/2+2pin, pi+2pin), n in Z`
`darrdarrdarr x in [-pi/2+2pin, 2pin)uu`
`uu(2pin, pi/2+2pin], n in Z`
Extrêmes
`y_min = 1, x = pi/2+2pin, n in Z`
`y_max = -1, x = -pi/2+2pin, n in Z`
Points d'inflections
`non`
`text(csc) alpha = 1/sin alpha`
`text(csc) alpha = (cot alpha)/cos alpha`
`text(csc) alpha = -sec(alpha+pi/2) = sec(alpha-pi/2)`
`-text(csc) alpha = text(csc)(-alpha)`
`(text(csc) alpha)^' = - cos alpha/sin^2 alpha`
`int (text(csc) alpha)dalpha = -ln |tan(alpha/2)| + C`