Section 5.5 Inverse Trigonometric Functions and Their Graphs
Section 5.5 Inverse Trigonometric Functions and Their Graphs
DEFINITION: The inverse sine function, denoted by sin-1 x (or arcsin x), is defined to be
the inverse of the restricted sine function
sin x, - x
2
2
DEFINITION: The inverse cosine function, denoted by cos-1 x (or arccos x), is defined to be the inverse of the restricted cosine function
cos x, 0 x
DEFINITION: The inverse tangent function, denoted by tan-1 x (or arctan x), is defined
to be the inverse of the restricted tangent function
tan x, - < x <
2
2
DEFINITION: The inverse cotangent function, denoted by cot-1 x (or arccot x), is defined to be the inverse of the restricted cotangent function
cot x, 0 < x <
1
DEFINITION: The inverse secant function, denoted by sec-1 x (or arcsec x), is defined to
be the inverse of the restricted secant function
[
]
sec x, x [0, /2) [, 3/2) or x [0, /2) (/2, ] in some other textbooks
DEFINITION: The inverse cosecant function, denoted by csc-1 x (or arccsc x), is defined
to be the inverse of the restricted cosecant function
[
]
csc x, x (0, /2] (, 3/2] or x [-/2, 0) (0, /2] in some other textbooks
IMPORTANT: Do not confuse sin-1 x, cos-1 x, tan-1 x, cot-1 x, sec-1 x, csc-1 x
with
1
1
1
1
1
1
,
,
,
,
,
sin x cos x tan x cot x sec x csc x
FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x
DOMAIN [-1, 1] [-1, 1]
(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)
RANGE [-/2, /2]
[0, ] (-/2, /2)
(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]
2
FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x
DOMAIN [-1, 1] [-1, 1]
(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)
RANGE [-/2, /2]
[0, ] (-/2, /2)
(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]
EXAMPLES:
(a)
sin-1 1
=
,
since
sin
=
1
and
[ -,
] .
2
2
2
22
(b)
sin-1(-1)
=
-,
since
( ) sin -
=
-1
and
-
[ -,
] .
2
2
2
22
(c)
sin-1
0
=
0,
since
sin
0
=
0
and
0
[ -,
] .
22
(d)
sin-1
1
=
,
since
sin
=
1
and
[ -,
] .
26
62 6
22
(e)
sin-1
3
=
,
since
sin
=
3
and
[ -,
] .
23
32
3
22
(f)
sin-1
2
=
,
since
sin
=
2
and
[ -,
] .
24
42
4
22
EXAMPLES:
cos-1
0
=
,
2
tan-1
1
=
,
4
cos-1 1 = 0,
cos-1(-1) = ,
cos-1
1
=
,
cos-1
3 =,
cos-1
2 =
23
26
24
tan-1(-1)
=
-
,
tan-1 3
=
,
tan-1 1
=,
()
tan-1 - 1
=-
4
3
36
3
6
EXAMPLES: Find sec-1 1, sec-1(-1), and sec-1(-2).
3
FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x
DOMAIN [-1, 1] [-1, 1]
(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)
RANGE [-/2, /2]
[0, ] (-/2, /2)
(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]
EXAMPLES: Find sec-1 1, sec-1(-1), and sec-1(-2).
Solution: We have sec-1 1 = 0,
sec-1(-1) = ,
sec-1(-2) = 4 3
since
sec 0 = 1,
sec = -1,
4 sec = -2
3
and
4 [ ) [ 3 )
0, , 0, ,
3
2
2
2 Note that sec 3 is also -2, but
sec-1(-2) = 2 3
since
2 [ ) [ 3 )
0, ,
3
2
2
EXAMPLES: Find
tan-1 0
cot-1 0
cot-1 1
sec-1 2
csc-1 2
csc-1 2 3
4
FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x
DOMAIN [-1, 1] [-1, 1]
(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)
RANGE [-/2, /2]
[0, ] (-/2, /2)
(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]
EXAMPLES: We have
tan-1 0 = 0,
cot-1
0
=
,
2
cot-1
1
=
,
4
sec-1
2
=
,
4
csc-1
2
=
,
6
csc-1 2
=
33
EXAMP(LES: Eva)luate ( )
(
)
7
(a) sin arcsin , arcsin sin , and arcsin sin .
6
6
6
(
)
( )
(
)
8
(b) sin arcsin , arcsin sin , and arcsin sin .
7
7
7
(
( ))
(
)
(
)
2
2
9
(c) cos arccos - , arccos cos , and arccos cos .
5
5
5
Solution: Since arcsin x is the inverse of the restricted sine function, we have
sin(arcsin x) = x if x [-1, 1] and arcsin(sin x) = x if x [-/2, /2]
Therefo(re
)
( )
(a) sin arcsin = arcsin sin = , but
6
66
(
)
()
7
1
arcsin sin = arcsin - = -
6
2
6
or
(
)
7
( ( ))
(
)
( )
arcsin sin = arcsin sin + = arcsin - sin = - arcsin sin = -
6
6
6
6
6
(
)
( )
(b) sin arcsin = arcsin sin = , but
7
77
(
)
8
( ( ))
(
)
( )
arcsin sin 7
= arcsin sin + 7
= arcsin
- sin 7
= - arcsin
sin 7
= -7
(c) Similarly, since arccos x is the inverse of the restricted cosine function, we have
cos(arccos x) = x if x [-1, 1] and arccos(cos x) = x if x [0, ]
(
( ))
(
)
2
2
2 2
Therefore cos arccos - 5
= - 5 and arccos
cos 5
= , but 5
(
)
9
((
))
( )
arccos cos = arccos cos 2 - = arccos cos =
5
5
55
5
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