1)
The notations
2)
Range
of usual principal value
3)
Definitions
as infinite series
4)
Logarithmic
forms
4.1
Natural logarithm's expressions
4.2 Logarithmic
formulas and connections
4.3 Logarithmic
formulas. Proofs
5)
Derivatives
of inverse trigonometric functions
6)
Indefinite
integrals of inverse trigonometric functions
1) The notations
For inverse trigonometric
functions, the notations sin-1
and cos-1
are often used for arcsin
and arccos,
etc. When this notation is used, the
inverse functions are sometimes confused
with the multiplicative inverses of
the functions. The notation using the
"arc-" prefix avoids such
confusion.
The inverse trigonometric
and hyperbolic functions are the multivalued
function that are the inverse functions
of the trigonometric and hyperbolic
functions.
Top
2) Range of usual principal value
The trigonometric functions
are periodic, so we must restrict their
domains before we are able to define
a unique inverse.
| y
= f(z) |
Alternate
notations |
Range
of usual principal value |
| y=sin-1(z) |
y=arcsin(z) |
 |
| y=cos-1(z) |
y=arccos(z) |
 |
| y=tan-1(z) |
y=arctan(z) |
 |
| y=cot-1(z) |
y=arccot(z) |
y 0
|
| y=sec-1(z) |
y=arcsec(z) |
y |
| y=csc-1(z) |
y=arccsc(z) |
y0
|
Top
3)
Definitions as
infinite series
The inverse trigonometric
functions can be defined in terms of
infinite series.
arcsin(z)=
arccos(z)=
- arcsin(z)
=
=
-
arctan(z)=
arccot(z)=
- arctan(z)
=
=
-
arccsc(z)
= arcsin(z
-1) =
= 
arcsec(z)
= arccos(z
-1) =
=
-
Top
4) Logarithmic forms
4.1
Natural logarithm's expressions
The inverse trigonometric
functions may be expressed using natural
logarithms.
arcsin(z)
= -i
ln(iz
+
)
arccos(z)
= -i
ln(z
+
)
arctan(z)
= 
(ln(1
- iz)
- (ln(1
+
iz))
arccot(z)
= (ln(1
- 
)
- (ln(1
+
))
arccsc(z)
= -i
ln(
+
)
arcsec(z)
= -i
ln(
+
)
Top
4.2
Logarithmic formulas and connections
Top
4.3 Logarithmic formulas and
connections. Proofs
Top
5) Derivatives
of inverse trigonometric functions
arcsin(z)
= 
arccos(z)
= -
arctan(z)
= 
arccot(z)
= -
arccsc(z)
= - 
arcsec(z)
=
Top
6) Indefinite
integrals of inverse trigonometric functions
arcsin(z)dz
= zarcsin(z)
+ +
C
arccos(z)dz
= zarccos(z)
- +
C
arctan(z)dz
= zarctan(z)
- 
ln(1+
z2)
+ C
arccot(z)dz
= zarccot(z)
+ ln(1+
z2)
+ C
arccsc(z)dz
= zarccsc(z)
+ ln(z
+ )
+ C
arcsec(z)dz
= zarcsec(z)
- ln(z
+ )
+ C
by
Tetyana Butler
