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arcsin
(
x
)
=
∫
0
x
1
1
−
t
2
d
t
=
−
i
log
(
i
x
+
1
−
x
2
)
{\displaystyle \arcsin(x)=\int _{0}^{x}{\frac {1}{\sqrt {1-t^{2}}}}\mathrm {d} t=-i\log(ix+{\sqrt {1-x^{2}}})}
arccos
(
x
)
=
π
2
−
arcsin
(
x
)
=
π
2
−
∫
0
x
1
1
−
t
2
d
t
=
π
2
+
i
log
(
i
x
+
1
−
x
2
)
{\displaystyle \arccos(x)={\frac {\pi }{2}}-\arcsin(x)={\frac {\pi }{2}}-\int _{0}^{x}{\frac {1}{\sqrt {1-t^{2}}}}\mathrm {d} t={\frac {\pi }{2}}+i\log(ix+{\sqrt {1-x^{2}}})}
arctan
(
x
)
=
∫
0
x
1
1
+
t
2
d
t
=
i
2
log
(
1
−
i
x
1
+
i
x
)
{\displaystyle \arctan(x)=\int _{0}^{x}{\frac {1}{1+t^{2}}}\mathrm {d} t={\frac {i}{2}}\log \left({\frac {1-ix}{1+ix}}\right)}
arccsc
(
x
)
=
arcsin
(
1
x
)
=
−
i
log
(
i
x
+
1
−
1
z
2
)
{\displaystyle \operatorname {arccsc}(x)=\arcsin \left({\frac {1}{x}}\right)=-i\log \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{z^{2}}}}}\right)}
arcsec
(
x
)
=
arccos
(
1
x
)
=
π
2
−
arcsin
(
1
x
)
=
π
2
+
i
log
(
i
x
+
1
−
1
z
2
)
{\displaystyle \operatorname {arcsec}(x)=\arccos \left({\frac {1}{x}}\right)={\frac {\pi }{2}}-\arcsin \left({\frac {1}{x}}\right)={\frac {\pi }{2}}+i\log \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{z^{2}}}}}\right)}
arccot
(
x
)
=
arctan
(
1
x
)
=
π
2
−
arctan
(
x
)
=
π
2
+
i
2
log
(
1
+
i
x
1
−
i
x
)
{\displaystyle \operatorname {arccot}(x)=\arctan \left({\frac {1}{x}}\right)={\frac {\pi }{2}}-\arctan(x)={\frac {\pi }{2}}+{\frac {i}{2}}\log \left({\frac {1+ix}{1-ix}}\right)}
arcsin
(
x
)
+
arcsin
(
y
)
=
arcsin
(
x
1
−
y
2
+
y
1
−
x
2
)
{\displaystyle \arcsin(x)+\arcsin(y)=\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}
arccos
(
x
)
+
arccos
(
y
)
=
arccos
(
x
y
−
(
1
−
x
2
)
(
1
−
y
2
)
)
{\displaystyle \arccos(x)+\arccos(y)=\arccos \left(xy-{\sqrt {(1-x^{2})(1-y^{2})}}\right)}
arctan
(
x
)
+
arctan
(
y
)
=
arctan
(
x
+
y
1
−
x
y
)
(
mod
π
)
{\displaystyle \arctan(x)+\arctan(y)=\arctan \left({\frac {x+y}{1-xy}}\right){\pmod {\pi }}}
