Overzicht van de cyclometrische functies
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Naam
Notatie
Definitie
Domein
Bereik
Boogsinus
y
=
arcsin
(
x
)
{\displaystyle y=\arcsin(x)}
y
=
b
g
s
i
n
(
x
)
{\displaystyle y=\mathrm {bgsin} (x)}
x
=
sin
(
y
)
{\displaystyle x=\sin(y)}
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
−π/2 ≤ y ≤ π/2
Boogcosinus
y
=
arccos
(
x
)
{\displaystyle y=\arccos(x)}
y
=
b
g
c
o
s
(
x
)
{\displaystyle y=\mathrm {bgcos} (x)}
x
=
cos
(
y
)
{\displaystyle x=\cos(y)}
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
0 ≤ y ≤ π
Boogtangens
y
=
arctan
(
x
)
{\displaystyle y=\arctan(x)}
y
=
b
g
t
a
n
(
x
)
{\displaystyle y=\mathrm {bgtan} (x)}
x
=
tan
(
y
)
{\displaystyle x=\tan(y)}
−
∞
≤
x
≤
∞
{\displaystyle -\infty \leq x\leq \infty }
−π/2 < y < π/2
Boogcotangens
y
=
arccot
(
x
)
{\displaystyle y=\operatorname {arccot}(x)}
y
=
b
g
c
o
t
(
x
)
{\displaystyle y=\mathrm {bgcot} (x)}
x
=
cot
(
y
)
{\displaystyle x=\cot(y)}
−
∞
<
x
<
∞
{\displaystyle -\infty <x<\infty }
0 < y < π
Boogsecans
y
=
arcsec
(
x
)
{\displaystyle y=\operatorname {arcsec}(x)}
y
=
b
g
s
e
c
(
x
)
{\displaystyle y=\mathrm {bgsec} (x)}
x
=
sec
(
y
)
{\displaystyle x=\sec(y)}
−
∞
<
x
<
−
1
{\displaystyle -\infty <x<-1}
of
1
<
x
<
∞
{\displaystyle 1<x<\infty }
0
≤
y
<
1
2
π
{\displaystyle 0\leq y<{\tfrac {1}{2}}\pi }
of
1
2
π
<
y
<
π
{\displaystyle {\tfrac {1}{2}}\pi <y<\pi }
Boogcosecans
y
=
arccsc
(
x
)
{\displaystyle y=\operatorname {arccsc}(x)}
y
=
b
g
c
s
c
(
x
)
{\displaystyle y=\mathrm {bgcsc} (x)}
x
=
csc
(
y
)
{\displaystyle x=\csc(y)}
−
∞
<
x
<
−
1
{\displaystyle -\infty <x<-1}
of
1
<
x
<
∞
{\displaystyle 1<x<\infty }
0
≤
y
<
1
2
π
{\displaystyle 0\leq y<{\tfrac {1}{2}}\pi }
of
1
2
π
<
y
<
π
{\displaystyle {\tfrac {1}{2}}\pi <y<\pi }
Logaritmische vormen
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Deze functies kunnen ook aan de hand van complexe logaritmen uitgedrukt worden:
arcsin
x
=
−
i
ln
(
i
x
+
1
−
x
2
)
arccos
x
=
−
i
ln
(
x
+
x
2
−
1
)
=
π
2
+
i
ln
(
i
x
+
1
−
x
2
)
arctan
x
=
i
2
(
ln
(
1
−
i
x
)
−
ln
(
1
+
i
x
)
)
arccot
x
=
i
2
(
ln
(
1
−
i
x
)
−
ln
(
1
+
i
x
)
)
arcsec
x
=
−
i
ln
(
1
x
2
−
1
+
1
x
)
=
i
ln
(
1
−
1
x
2
+
i
x
)
+
π
2
arccsc
x
=
−
i
ln
(
1
−
1
x
2
+
i
x
)
{\displaystyle {\begin{aligned}\arcsin x&{}=-i\,\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)\\\arccos x&{}=-i\,\ln \left(x+{\sqrt {x^{2}-1}}\right)={\frac {\pi }{2}}\,+i\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)\\\arctan x&{}={\frac {i}{2}}\left(\ln \left(1-i\,x\right)-\ln \left(1+i\,x\right)\right)\\\operatorname {arccot} x&{}={\frac {i}{2}}\left(\ln \left(1-{\frac {i}{x}}\right)-\ln \left(1+{\frac {i}{x}}\right)\right)\\\operatorname {arcsec} x&{}=-i\,\ln \left({\sqrt {{\frac {1}{x^{2}}}-1}}+{\frac {1}{x}}\right)=i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)+{\frac {\pi }{2}}\\\operatorname {arccsc} x&{}=-i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)\end{aligned}}}
Goniometrische gelijkheden
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