Since arcsin(
x
) is defined only for

1
≤
x
≤
1, the equivalence cos (2 arcsin(
x
)) = 1

2
x
2
is valid only on [

1
,
1].
A few remarks about Example
10.6.1
are in order.
Most of the common errors encountered in
dealing with the inverse circular functions come from the need to restrict the domains of the
original functions so that they are onetoone. One instance of this phenomenon is the fact that
arccos
(
cos
(
11
π
6
))
=
π
6
as opposed to
11
π
6
. This is the exact same phenomenon discussed in Section
5.2
when we saw
p
(

2)
2
= 2 as opposed to

2.
Additionally, even though the expression we
arrived at in part
2b
above, namely 1

2
x
2
, is defined for all real numbers, the equivalence
cos (2 arcsin(
x
)) = 1

2
x
2
is valid for only

1
≤
x
≤
1. This is akin to the fact that while the
expression
x
is defined for all real numbers, the equivalence (
√
x
)
2
=
x
is valid only for
x
≥
0. For
this reason, it pays to be careful when we determine the intervals where such equivalences are valid.
The next pair of functions we wish to discuss are the inverses of tangent and cotangent, which
are named arctangent and arccotangent, respectively.
First, we restrict
f
(
x
) = tan(
x
) to its
fundamental cycle on
(

π
2
,
π
2
)
to obtain
f

1
(
x
) = arctan(
x
). Among other things, note that the
vertical
asymptotes
x
=

π
2
and
x
=
π
2
of the graph of
f
(
x
) = tan(
x
) become the
horizontal
asymptotes
y
=

π
2
and
y
=
π
2
of the graph of
f

1
(
x
) = arctan(
x
).
x
y

π
4

π
2
π
4
π
2

1
1
f
(
x
) = tan(
x
),

π
2
< x <
π
2
.
reflect across
y
=
x
→
switch
x
and
y
coordinates
x
y

π
4

π
2
π
4
π
2

1
1
f

1
(
x
) = arctan(
x
).
Next, we restrict
g
(
x
) = cot(
x
) to its fundamental cycle on (0
, π
) to obtain
g

1
(
x
) = arccot(
x
).
Once again, the vertical asymptotes
x
= 0 and
x
=
π
of the graph of
g
(
x
) = cot(
x
) become the
horizontal asymptotes
y
= 0 and
y
=
π
of the graph of
g

1
(
x
) = arccot(
x
). We show these graphs
on the next page and list some of the basic properties of the arctangent and arccotangent functions.
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824
Foundations of Trigonometry
x
y
π
4
π
2
3
π
4
π

1
1
g
(
x
) = cot(
x
), 0
< x < π
.
reflect across
y
=
x
→
switch
x
and
y
coordinates
x
y
π
4
π
2
3
π
4
π

1
1
g

1
(
x
) = arccot(
x
).
Theorem 10.27. Properties of the Arctangent and Arccotangent Functions
Properties of
F
(
x
) = arctan(
x
)
–
Domain: (
∞
,
∞
)
–
Range:
(

π
2
,
π
2
)
–
as
x
→ ∞
, arctan(
x
)
→ 
π
2
+
; as
x
→ ∞
, arctan(
x
)
→
π
2

–
arctan(
x
) =
t
if and only if

π
2
< t <
π
2
and tan(
t
) =
x
–
arctan(
x
) = arccot
(
1
x
)
for
x >
0
–
tan (arctan(
x
)) =
x
for all real numbers
x
–
arctan(tan(
x
)) =
x
provided

π
2
< x <
π
2
–
additionally, arctangent is odd
Properties of
G
(
x
) = arccot(
x
)
–
Domain: (
∞
,
∞
)
–
Range: (0
, π
)
–
as
x
→ ∞
, arccot(
x
)
→
π

; as
x
→ ∞
, arccot(
x
)
→
0
+
–
arccot(
x
) =
t
if and only if 0
< t < π
and cot(
t
) =
x
–
arccot(
x
) = arctan
(
1
x
)
for
x >
0
–
cot (arccot(
x
)) =
x
for all real numbers
x
–
arccot(cot(
x
)) =
x
provided 0
< x < π
10.6 The Inverse Trigonometric Functions
825
Example 10.6.2.