Calculus
Grinshpan
INVERSE TRIGONOMETRIC FUNCTIONS
There are three most common inverse trigonometric functions: inverse
sine, inverse cosine, and inverse tangent.
1. Inverse sine function
The inverse sine function is written as y = sin−1 (x) or y = arcsin x.
(Not to be confused with y = 1/ sin x.)
The domain of arcsin x is the interval [−1, 1] and its range is [− π2 , π2 ].
For any number x between −1 and 1, arcsin x is the angle
between − π2 and π2 whose sine is x.
In other words, let −1 ≤ x ≤ 1 and let − π2 ≤ θ ≤ π2 . Then to say that
arcsin x = θ
is the same as to say that
sin θ = x.
Here are some examples:
arcsin 0 = 0
1
= π6
arcsin
2
√
arcsin(− 23 ) = − π3
arcsin 1 = π2
because
because
because
because
sin 0 = 0
sin π6 = 12
√
sin(− π3 ) = − 23
sin π2 = 1.
By the definition of arcsin we have
arcsin(sin θ) = θ,
for any − π2 ≤ θ ≤ π2 ,
sin(arcsin x) = x,
for any −1 ≤ x ≤ 1.
and
More examples. First, sin π = 0 but arcsin 0 6= π because π does not
belong to [− π2 , π2 ]. Second, arcsin π3 is not defined because π3 does not
belong to [−1, 1].
1
2
2. Inverse cosine function
The inverse cosine function is written as y = cos−1 (x) or y = arccos x.
(Not to be confused with y = 1/ cos x.)
The domain of arccos x is the interval [−1, 1] and its range is [0, π].
For any number x between −1 and 1, arccos x is the angle
between 0 and π whose cosine is x.
In other words, let −1 ≤ x ≤ 1 and let 0 ≤ θ ≤ π. Then to say that
arccos x = θ
is the same as to say that
cos θ = x.
Here are some examples:
arccos 0 = π2
1
arccos
= π3
√ 2
arccos(− 23 ) = 5π
6
arccos 1 = 0
because
because
because
because
cos π2 = 0
cos π3 = 12 √
cos( 5π
) = − 23
6
cos 0 = 1.
By the definition of arccos we have
arccos(cos θ) = θ,
for any 0 ≤ θ ≤ π,
and
cos(arccos x) = x,
for any −1 ≤ x ≤ 1.
More examples. First, cos(2π) = 1 but arccos 1 6= 2π because 2π does
not belong to [0, π]. Second, arccos 5 is not defined because 5 does not
belong to [−1, 1].
3
3. Inverse tangent function
The inverse tangent function is written as y = tan−1 (x) or
y = arctan x. (Not to be confused with y = 1/ tan x = cot x.)
The domain of arctan x is the entire real line (−∞, ∞) and its range
is (− π2 , π2 ).
For any number x, arctan x is the angle between − π2 and
whose tangent is x.
π
2
In other words, let − π2 < θ < π2 . Then to say that
arctan x = θ
is the same as to say that
tan θ = x.
Here are some examples:
arctan 0 = 0
arctan 1 = π4
√
arctan(− 3) = − π3
arctan √13 = π6
because
because
because
because
tan 0 = 0
tan π4 = 1
√
tan(− π3 ) = − 3
tan π6 = √13 .
By the definition of arctan we have
arctan(tan θ) = θ,
for any − π2 < θ < π2 ,
and
tan(arctan x) = x,
for any x.
More examples. First, tan π = 0 but arctan 0 6= π because π does not
belong to (− π2 , π2 ). Second, arctan x is defined for any x.
4
4. Less common inverse trigonometric functions
We can also consider inverse cotangent, inverse secant, and inverse
cosecant. These functions will not appear so often in the course and
the information below is just for your reference.
Inverse cotangent. Written as y = cot−1 x or y =arccot(x). Given a
number −∞ < x < ∞, arccot(x) returns the angle 0 < θ < π whose
cotangent is x. For instance, arccot(0) = π2 , arccot(1) = π4 ,
√
,
arccot(−
.
3) = 5π
arccot(−1) = 3π
4
6
Inverse secant. Written as y = sec−1 x or y =arcsec(x). Given a
number x with |x| ≥ 1, arcsec(x) returns the angle 0 ≤ θ ≤ π, θ 6= π2 ,
whose secant is x. The reason why π2 is not in the range is that sec π2
is undefined. Examples: arcsec(1) = 0, arcsec(2) = π3 , arcsec(−1) = π,
arcsec(− √23 ) = 5π
.
6
Inverse cosecant. Written as y = csc−1 x or y =arccsc(x). Given a
number x with |x| ≥ 1, arccsc(x) returns the angle − π2 ≤ θ ≤ π2 ,
θ 6= 0, whose cosecant is x. The reason why 0 is not in the range is
that csc 0 is undefined. Examples: arccsc(1) = π2 , arccsc(2) = π6 ,
arccsc(−1) = − π2 , arccsc(− √23 ) = − π3 .