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Section 5.6 Inverse Trigonometric Functions: Differentiation
* None of the six basic trigonometric functions has an inverse function
( they are not one-to-one ) , unless you fix / restrict their domains.
y = sin x
y = sin x
not one-to-one
one-to-one with
restricted domain
Domain: [ −π 2, π 2]
Range: [ −1,1]
Domain of y =
sin x ⇒ Range of y =
arcsin x
Range of y =
sin x ⇒ Domain of y =
arcsin x
Domain: [ −1,1]
reflect restricted
sine function
about y = x
=
y arcsin
=
x sin −1 x
=
y arcsin
=
x iff sin y x
Domain: [ −1,1]
Range: [ −π 2, π 2]
Range: [ −π 2, π 2]
y = cos x
y = cos x
one-to-one with
restricted domain
not one-to-one
Domain: [ 0, π ]
Range: [ −1,1]
Domain: [ −1,1]
reflect restricted
cosine function
about y = x
=
y arccos
=
x cos −1 x
=
y arccos
=
x iff cos y x
Domain: [ −1,1]
Range: [ 0, π ]
Range: [ 0, π ]
y = tan x
vertical asymptote
y = tan x
not one-to-one
one-to-one with
restricted domain
=
y arctan
=
x tan −1 x
Domain: ( −π 2, π 2 )
Range: ( −∞, ∞ )
Domain: ( −∞, ∞ )
Range: ( −π 2, π 2 )
reflect restricted
tangent function
about y = x
=
y arctan
=
x iff tan y x
Domain: ( −∞, ∞ )
Range: ( −π 2, π 2 )
horizontal asymptote
y = cot x
not one-to-one
y = cot x
Domain: ( 0, π )
one-to-one with
restricted domain
reflect restricted
cotangent function
about y = x
Range: ( −∞, ∞ )
−1
=
y arc
=
cot x cot x
=
y arc
=
cot x iff cot y x
Domain: ( −∞, ∞ )
Range: ( 0, π )
Domain: ( −∞, ∞ )
Range: ( 0, π )
y = sec x
not one-to-one
y = sec x
one-to-one with
restricted domain
Domain: ( 0, π ) x ≠ π 2
Range: ( −∞, −1]  [1, ∞ )
y arc
sec x sec −1 x
=
=
reflect restricted
secant function
about y = x
Domain: ( −∞, −1]  [1, ∞ )
Range: ( 0, π ) y ≠ π 2
=
y arc
=
sec x iff sec y x
Domain: ( −∞, −1]  [1, ∞ )
Range: ( 0, π ) y ≠ π 2
y = csc x
not one-to-one
y = csc x
one-to-one with
restricted domain
Domain: ( −π 2, π 2 ) x ≠ 0
Range: ( −∞, −1]  [1, ∞ )
=
y arccsc
=
x csc −1 x
reflect restricted
cosecant function
about y = x
Domain: ( −∞, −1]  [1, ∞ )
Range: ( −π 2, π 2 ) y ≠ 0
=
y arc
=
csc x iff csc y x
Domain: ( −∞, −1]  [1, ∞ )
Range: ( −π 2, π 2 ) y ≠ 0
Definitions of Inverse Trigonometric Functions
Range
Domain
Function
y arcsin
=
x iff sin y x
y arccos
=
x iff cos y x
y arctan
=
x iff tan y x
−1 ≤ x ≤ 1
−1 ≤ x ≤ 1
−∞ < x < ∞
arc
=
cot x iff cot y x
−∞ < x < ∞
y
y arc
=
sec x iff sec y x
y arc
=
csc x iff csc y x
x ≥1
x ≥1
−π 2 ≤ y ≤ π 2
0≤ y ≤π
−π 2 < y < π 2
0< y <π
0 ≤ y ≤π, y ≠ π 2
−π 2 ≤ y ≤ π 2, y ≠ 0
Inverse Relationship Between Trigonometric Functions and Inverse Trigonometric Functions
If − 1 ≤ x ≤ 1 and −π 2 ≤ y ≤ π 2, then
=
sin ( arcsin x ) x and
=
arcsin ( sin y ) y
If − 1 ≤ x ≤ 1 and 0 ≤ y ≤ π , then
=
cos ( arccos x ) x and
=
arccos ( cos y ) y
=
=
If −π 2 < y < π 2, then
tan ( arctan x ) x and
arctan ( tan y ) y
If 0 < y < π , then
=
cot ( arc cot x ) x and
=
arc cot ( cot y ) y
If x ≥ 1 and 0 ≤ y < π 2 or π 2 < y ≤ π , then
=
sec ( arc sec x ) x and
=
arc sec ( sec y ) y
If x ≥ 1 and −π 2 ≤ y < 0 or 0 < y < π 2, then
=
csc ( arc csc x ) x and
=
arc csc ( csc y ) y
Derivatives of Inverse Trigonometric Functions
d
u′
arcsin
u
=
[
]
dx
1− u2
d
−u ′
arccos
u
=
[
]
dx
1− u2
d
u′
[arctan u ] =
dx
1+ u2
−u ′
d
[arc cot u ] =
1+ u2
dx
d
u′
[arc sec u ] =
dx
u u2 −1
d
−u ′
[arc csc u ] =
dx
u u2 −1