Inverse trigonometric functions
Inverse functions. If a function f maps a subset D of its domain one-to-one onto its range, the restriction of
f to D has an inverse, f −1 , whose domain is the range of f and whose range is D. By definition,
x = f −1 (y)
f (x) = y
means that
x ∈ D.
and
(†)
It follows that f (f −1 (y)) = y for all y ∈ dom(f −1 ), and that f −1 (f (x)) = x if, and only if, x ∈ D.
The inverse secant function. The restriction of the secant function to [ 0, 12 π ) ∪ [ π, 32 π ) is one-to-one; its
inverse is called the inverse secant, or arcsecant, function, and is denoted by arcsec. The domain of arcsec is
( −∞, −1 ] ∪ [ 1, ∞ ) and the range of arcsec is [ 0, 12 π ) ∪ [ π, 32 π ). Below is a graph of z = sec ϑ, with its
values on [ 0, 21 π ) ∪ [ π, 32 π ) emphasized, and beside it the graph of ϑ = arcsec z.
z = sec ϑ
The inverse sine function. The restriction of the sine function to [ − 12 π, 12 π ] is one-to-one; its inverse is
called the inverse sine, or arcsine, function, and is denoted by arcsin. The domain of arcsin is [ −1, 1 ] and
the range of arcsin is [ − 12 π, 12 π ]. Below is a graph of y = sin ϑ, with its values on [ − 12 π, 21 π ] emphasized,
and beside it the graph of ϑ = arcsin y.
ϑ = arcsec z
3
π
2
ϑ=
1
π
2
π
ϑ = arcsin y
y = sin ϑ
ϑ=
1
π
2
π
−π
ϑ
1
1
π
2
− 12 π
ϑ
1 y
−1
−1
z
1
−1
By definition,
− 12 π
By definition,
ϑ = arcsec z
ϑ = arcsin y
means that
sin ϑ = y
and
− 12 π 6 ϑ 6
1
π.
2
for
sec ϑ = z
and
06ϑ<
1
π
2
or
π6ϑ<
− 1 < y < 1.
Observe that the graph of ϑ = arcsin y has (one-sided) vertical tangents where y = ±1. Other symbols for
the inverse sine function include asin and sin−1 .
The other inverse trigonometric functions. The remaining inverse trigonometric functions are defined using
the cofunction identities for the corresponding trigonometric functions. Below are their definitions and graphs.
Their domains, ranges and derivatives can be seen from their definitions (and/or their graphs).
arccos x =
( − 12 π, 12 π )
The inverse tangent function. The restriction of the tangent function to
is one-to-one, and its
inverse is called the inverse tangent, or arctangent, function, which is denoted by arctan. The domain of
arctan is R and the range of arctan is ( − 12 π, 12 π ). Below is a graph of y = tan ϑ, with its values on
( − 21 π, 12 π ) emphasized, and beside it the graph of ϑ = arctan y.
1
π
2
− arcsin x
arccot x =
ϑ = arccos x
1
π
2
− arctan x
ϑ = arccot x
π
ϑ=π
t = tan ϑ
1
π
2
x
−1
−π
π
x
ϑ
1
π
2
1
π
2
ϑ = arctan t
ϑ=
1 x
arccsc x =
1
π
2
− arcsec x
ϑ = arccsc x
ϑ = − 21 π
1
π
2
By definition,
ϑ = arctan t
means that
tan ϑ = t
and
− 12 π < ϑ <
As before, one can use the equation tan ϑ = t, and the restriction on ϑ, to show that
d
1
(arctan t) =
,
for
t ∈ R.
dt
1 + t2
Other symbols for the inverse tangent function include atan and tan−1 .
3
π.
2
Again, one can use the equation sec ϑ = z and the restriction in ϑ, to show that
1
d
(arcsec z) = √
,
for
|z| > 1.
dz
z z2 − 1
Observe that the graph of ϑ = arcsec z has (one-sided) vertical tangents where z = ±1. Other symbols for
the inverse secant function include asec and sec−1 .
Differentiating the equation sin ϑ = y implicitly with respect to y gives
dϑ
1
dϑ
= 1,
or
=
,
provided
− 12 π < ϑ < 12 π.
cos ϑ
dy
dy
cos ϑ
p
p
Since cos ϑ > 0 if − 21 π < ϑ < 12 π, it follows that cos ϑ = 1 − sin2 ϑ = 1 − y 2 . Therefore,
d
1
,
(arcsin y) = p
dy
1 − y2
means that
x
1
π.
2
− 21 π
ϑ=π