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 π π 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 π
≤ ≤
DEFINITION: The inverse inverse tangent function, denoted by tan−1 x (or arctan x), is defined to be the inverse of the restricted tangent function π π tan x,
−
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 < π
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
�or x ∈ [0, π/2) ∪ (π/2, π] in some other textbooks�
∈ [0, π/2) ∪ [π, 3π/2)
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, π/2] ∪ (π, 3π/2]
�or x ∈ [−π/2, 0) ∪ (0, π/2] in some other textbooks�
IMPORTANT: Do not confuse sin−1 x, with
cos−1 x,
1 , sin x
FUNCTION sin−1 x cos−1 x tan−1 x cot−1 x sec−1 x csc−1 x
1 , cos x
tan−1 x,
cot−1 x,
1 , tan x
1 , cot x
DOMAIN [ 1, 1] [ 1, 1] ( ,+ ) ( ,+ ) , 1] [1, + ) , 1] [1, + )
− − −∞ ∞ −∞ ∞ (−∞ − ∪ ∞ (−∞ − ∪ ∞
2
sec−1 x, 1 , sec x
csc−1 x
1 csc x
RANGE [ π/2, π/2] [0, π] ( π/2, π/2) (0, π) [0, π/2) [π, 3π/2) (0, π/2] (π, 3π/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 [ π/2, π/2] [0, π] ( π/2, π/2) (0, π) [0, π/2) [π, 3π/2) (0, π/2] (π, 3π/2]
− − −∞ ∞ −∞ ∞ (−∞ − ∪ ∞ (−∞ − ∪ ∞
− −
∪ ∪
EXAMPLES: −1
(a) sin
−1
(b) sin
(c) sin−1
π π π 1 = , since sin = 1 and 2 2 2
−1
(e) sin
−1
(f) sin
2 2
π , since sin 2
� π� π � π π � (−1) = − − 2 = −1 and − 2 ∈ − 2 , 2 . � π π � 0 = 0, since sin 0 = 0 and 0 ∈ − , . 2 2
π π 1 π = , since sin = and 2 6 6 2 6
−1 1
(d) sin
� π π � ∈ − , .
√ 3
� π π � ∈ − , . 2 2
√ 3
π π π = , since sin = and 2 3 3 2 3
√ 2
√
π π 2 π = , since sin = and 2 4 4 2 4
� π π � ∈ − , . 2 2
� π π � ∈ − , . 2 2
EXAMPLES: π cos−1 0 = , 2 tan−1 1 =
π , 4
cos−1 1 = 0, tan−1 ( 1) =
−
1 π cos−1 = , 2 3
cos−1 ( 1) = π,
−
− π4 ,
tan−1
√
3=
π , 3
tan−1
EXAMPLES: Find sec−1 1, sec−1 ( 1), and sec−1 ( 2).
−
−
3
cos−1
√ 13 = π6 ,
√ 3
π = , 2 6 tan−1
cos−1
� 1� − √ 3
√ 2 2 =
=
− π6
π 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 [ π/2, π/2] [0, π] ( π/2, π/2) (0, π) [0, π/2) [π, 3π/2) (0, π/2] (π, 3π/2]
− − −∞ ∞ −∞ ∞ (−∞ − ∪ ∞ (−∞ − ∪ ∞
− −
∪ ∪
EXAMPLES: Find sec−1 1, sec−1 ( 1), and sec−1 ( 2).
−
Solution: We have sec−1 1 = 0,
−
sec−1 ( 1) = π,
−
sec−1 ( 2) =
−
4π 3
4π = 3
−2
since sec π =
sec 0 = 1, and
4π 0, π, 3
2π Note that sec is also 3
−1,
� π � � 3π � ∈ 0, ∪ π, 2
2
−2, but sec−1 ( 2) =
−
since
2π 3
sec
2π 3
� π � � 3π � ∈ 0, ∪ π, 2
2
EXAMPLES: Find tan−1 0
cot−1 0
cot−1 1
sec−1
4
√
2
csc−1 2
csc−1
√ 23
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 [ π/2, π/2] [0, π] ( π/2, π/2) (0, π) [0, π/2) [π, 3π/2) (0, π/2] (π, 3π/2]
− − −∞ ∞ −∞ ∞ (−∞ − ∪ ∞ (−∞ − ∪ ∞
− −
∪ ∪
EXAMPLES: We have −1
tan
−1
0 = 0,
cot
π 0= , 2
−1
cot
π 1= , 4
−1
sec
√
2=
π , 4
csc−1 2 =
π , 6
csc−1
√ 23 = π3
EXAMPLES: Evaluate π π 7π (a) sin arcsin , arcsin sin , and arcsin sin 6 6 6
� � � � � � . � 8π � � � π � π � (b) sin arcsin , arcsin sin , and arcsin sin . 7 7 7 � � 2 �� � 2π � � (c) cos arccos
−5
, arccos cos
9π , and arccos cos 5
5
�
.
Solution: Since arcsin x is the inverse of the restricted sine function, we have sin(arcsin x) = x if x
∈ [−1, 1]
arcsin(sin x) = x if x
and
∈ [−π/2, π/2]
Therefore
� � π � π � π (a) sin arcsin = arcsin sin = , but 6 6 6 � 7π � arcsin sin
6
= arcsin
� 1� −2
=
− π6
or
�
�
7π π arcsin sin = arcsin sin π + 6 6
� � �� = arcsin �− sin π � = − arcsin �sin π � = − π 6 6 6 � � π � π � π (b) sin arcsin = arcsin sin = , but 7 7 7 � 8π � � � �� � � π π � π � π arcsin sin = arcsin sin + π = arcsin − sin = − arcsin sin = − 7
7
7
7
(c) Similarly, since arccos x is the inverse of the restricted cosine function, we have cos(arccos x) = x if x Therefore cos
�
∈ [−1, 1]
arccos(cosx) = x if x
and
� 2 �� 2 � 2π � 2π arccos − = − and arccos cos = , but 5 5 5 5 � 9π � � � π �� � π �
arccos cos
5
= arccos cos 2π 5
−5
= arccos cos
5
∈ [0, π]
=
π 5
7
