c
Math 151, Spring 2010, Benjamin
Aurispa
4.6 Inverse Trigonometric Functions
Remember that only one-to-one functions have inverses. So, in order to find the inverse functions for sine,
cosine, and tangent, we must restrict their domains to intervals where they are one-to-one.
To find the inverse sine function, we restrict the domain of sine to [−π/2, π/2].
We define the inverse sine function, sin−1 x by sin−1 x = y ↔ sin y = x.
sin−1 x has domain
and range
.
sin−1 x is the ANGLE in the interval [−π/2, π/2] whose sine is x.
Remember that for inverse functions, we have “cancellation” laws. So, for the sine function
sin(sin−1 x) = x for −1 ≤ x ≤ 1
sin−1 (sin x) = x for − π2 ≤ x ≤
π
2
Examples:
sin−1
√
3
2
sin(sin−1
arcsin(− 12 )
9
10 )
sin−1 2
sin−1 (sin 4π
3 )
1
arcsin(sin π5 )
c
Math 151, Spring 2010, Benjamin
Aurispa
In order to have an inverse for cosine, we restrict the domain of cosine to the interval [0, π].
The inverse cosine function cos−1 is defined by cos−1 x = y ↔ cos y = x.
cos−1 x has domain
and range
.
cos−1 x is the ANGLE in the interval [0, π] whose cosine is x.
Cancellation Laws.
cos(cos−1 x) = x for −1 ≤ x ≤ 1
cos−1 (cos x) = x for 0 ≤ x ≤ π
Examples:
cos−1 (−
arccos(0)
cos(cos−1
12
11 )
√
2
2 )
cos(cos−1 (− 71 ))
cos−1 (cos 7π
6 )
2
arccos(cos 3π
4 )
c
Math 151, Spring 2010, Benjamin
Aurispa
In order to have an inverse for tangent, we restrict the domain of tangent to the interval (−π/2, π/2).
The inverse tangent function tan−1 is defined by tan−1 x = y ↔ tan y = x.
tan−1 x has domain
lim arctan x =
x→∞
and range
.
lim arctan x =
x→−∞
tan−1 x is the ANGLE in the interval (−π/2, π/2) whose tangent is x.
Cancellation Laws.
tan(tan−1 x) = x for all x
tan−1 (tan x) = x for − π2 < x <
Examples:
√
tan−1 (− 3)
arctan(1)
π
))
tan−1 (tan(− 12
arctan(tan( 5π
4 ))
3
π
2
tan(arctan 1000)
c
Math 151, Spring 2010, Benjamin
Aurispa
Examples: Evaluate the following expressions.
tan(sin−1 45 )
sin(2 cos−1 23 )
sin(tan−1 x)
cot(cos−1 x)
Derivatives of Inverse Trig Functions
d
1
sin−1 x = √
dx
1 − x2
d
1
cos−1 x = − √
dx
1 − x2
d
1
tan−1 x =
dx
1 + x2
d
1
sec−1 x = √
dx
x x2 − 1
4
c
Math 151, Spring 2010, Benjamin
Aurispa
Find the derivatives of the following functions.
f (x) = arcsin(3x2 + ln x)
y = x2 arctan(sec−1 x)
√
g(t) = (cos−1 ( t2 + sin t))−2
5