Chapter 2
INVERSE TRIGONOMETRIC
FUNCTIONS
2.1 Overview
2.1.1 Inverse function
Inverse of a function ‘f ’ exists, if the function is one-one and onto, i.e, bijective.
Since trigonometric functions are many-one over their domains, we restrict their
domains and co-domains in order to make them one-one and onto and then find
their inverse. The domains and ranges (principal value branches) of inverse
trigonometric functions are given below:
Functions Domain Range (Principal value
branches)
–π π
y = sin–1x [–1,1] 2 , 2
y = cos–1x [–1,1] [0,π]
–π π
y = cosec–1x R– (–1,1) 2 , 2 – {0}
π
y = sec–1x R– (–1,1) [0,π] –
2
–π π
y = tan–1x R ,
2 2
y = cot–1x R (0,π)
Notes:
(i) The symbol sin–1x should not be confused with (sinx)–1. Infact sin–1x is an
angle, the value of whose sine is x, similarly for other trigonometric functions.
(ii) The smallest numerical value, either positive or negative, of θ is called the
principal value of the function.
, INVERSE TRIGONOMETRIC FUNCTIONS 19
(iii) Whenever no branch of an inverse trigonometric function is mentioned, we mean
the principal value branch. The value of the inverse trigonometic function which
lies in the range of principal branch is its principal value.
2.1.2 Graph of an inverse trigonometric function
The graph of an inverse trigonometric function can be obtained from the graph of
original function by interchanging x-axis and y-axis, i.e, if (a, b) is a point on the graph
of trigonometric function, then (b, a) becomes the corresponding point on the graph of
its inverse trigonometric function.
It can be shown that the graph of an inverse function can be obtained from the
corresponding graph of original function as a mirror image (i.e., reflection) along the
line y = x.
2.1.3 Properties of inverse trigonometric functions
–
1. sin–1 (sin x) = x : x ∈ ,
2 2
cos–1(cos x) = x : x ∈[0, ]
–π π
tan–1(tan x) = x : x ∈ ,
2 2
cot–1(cot x) = x : x ∈ ( 0, π )
π
sec–1(sec x) = x : x ∈[0, π] –
2
–π π
cosec–1(cosec x) = x : x ∈ , – {0}
2 2
2. sin (sin–1 x) = x : x ∈[–1,1]
cos (cos–1 x) = x : x ∈[–1,1]
tan (tan–1 x) = x : x ∈R
cot (cot–1 x) = x : x ∈R
sec (sec–1 x) = x : x ∈R – (–1,1)
cosec (cosec–1 x) = x : x ∈R – (–1,1)
1
3. sin –1 = cosec –1 x : x ∈R – (–1,1)
x
1
cos –1 = sec –1 x : x ∈R – (–1,1)
x
, 20 MATHEMATICS
1
tan –1 = cot –1 x : x>0
x
= – π + cot–1x : x<0
4. sin–1 (–x) = –sin–1x : x ∈[–1,1]
cos–1 (–x) = π−cos–1x : x ∈[–1,1]
tan–1 (–x) = –tan–1x : x ∈R
cot–1 (–x) = π–cot–1x : x ∈R
sec–1 (–x) = π–sec–1x : x ∈R –(–1,1)
cosec–1 (–x) = –cosec–1x : x ∈R –(–1,1)
π
5. sin–1x + cos–1x = : x ∈[–1,1]
2
π
tan–1x + cot–1x = : x ∈R
2
π
sec–1x + cosec–1x = : x ∈R–[–1,1]
2
x+ y
6. tan–1x + tan–1y = tan–1 1 – xy : xy < 1
x− y
; xy > –1
tan–1x – tan–1y = tan–1 1 + xy
2x
7. 2tan–1x = sin–1 : –1 ≤ x ≤ 1
1 + x2
1 – x2
2tan–1x = cos–1 : x≥0
1 + x2
2x
2tan–1x = tan–1 : –1 < x < 1
1 – x2
2.2 Solved Examples
Short Answer (S.A.)
3
Example 1 Find the principal value of cos–1x, for x = .
2
INVERSE TRIGONOMETRIC
FUNCTIONS
2.1 Overview
2.1.1 Inverse function
Inverse of a function ‘f ’ exists, if the function is one-one and onto, i.e, bijective.
Since trigonometric functions are many-one over their domains, we restrict their
domains and co-domains in order to make them one-one and onto and then find
their inverse. The domains and ranges (principal value branches) of inverse
trigonometric functions are given below:
Functions Domain Range (Principal value
branches)
–π π
y = sin–1x [–1,1] 2 , 2
y = cos–1x [–1,1] [0,π]
–π π
y = cosec–1x R– (–1,1) 2 , 2 – {0}
π
y = sec–1x R– (–1,1) [0,π] –
2
–π π
y = tan–1x R ,
2 2
y = cot–1x R (0,π)
Notes:
(i) The symbol sin–1x should not be confused with (sinx)–1. Infact sin–1x is an
angle, the value of whose sine is x, similarly for other trigonometric functions.
(ii) The smallest numerical value, either positive or negative, of θ is called the
principal value of the function.
, INVERSE TRIGONOMETRIC FUNCTIONS 19
(iii) Whenever no branch of an inverse trigonometric function is mentioned, we mean
the principal value branch. The value of the inverse trigonometic function which
lies in the range of principal branch is its principal value.
2.1.2 Graph of an inverse trigonometric function
The graph of an inverse trigonometric function can be obtained from the graph of
original function by interchanging x-axis and y-axis, i.e, if (a, b) is a point on the graph
of trigonometric function, then (b, a) becomes the corresponding point on the graph of
its inverse trigonometric function.
It can be shown that the graph of an inverse function can be obtained from the
corresponding graph of original function as a mirror image (i.e., reflection) along the
line y = x.
2.1.3 Properties of inverse trigonometric functions
–
1. sin–1 (sin x) = x : x ∈ ,
2 2
cos–1(cos x) = x : x ∈[0, ]
–π π
tan–1(tan x) = x : x ∈ ,
2 2
cot–1(cot x) = x : x ∈ ( 0, π )
π
sec–1(sec x) = x : x ∈[0, π] –
2
–π π
cosec–1(cosec x) = x : x ∈ , – {0}
2 2
2. sin (sin–1 x) = x : x ∈[–1,1]
cos (cos–1 x) = x : x ∈[–1,1]
tan (tan–1 x) = x : x ∈R
cot (cot–1 x) = x : x ∈R
sec (sec–1 x) = x : x ∈R – (–1,1)
cosec (cosec–1 x) = x : x ∈R – (–1,1)
1
3. sin –1 = cosec –1 x : x ∈R – (–1,1)
x
1
cos –1 = sec –1 x : x ∈R – (–1,1)
x
, 20 MATHEMATICS
1
tan –1 = cot –1 x : x>0
x
= – π + cot–1x : x<0
4. sin–1 (–x) = –sin–1x : x ∈[–1,1]
cos–1 (–x) = π−cos–1x : x ∈[–1,1]
tan–1 (–x) = –tan–1x : x ∈R
cot–1 (–x) = π–cot–1x : x ∈R
sec–1 (–x) = π–sec–1x : x ∈R –(–1,1)
cosec–1 (–x) = –cosec–1x : x ∈R –(–1,1)
π
5. sin–1x + cos–1x = : x ∈[–1,1]
2
π
tan–1x + cot–1x = : x ∈R
2
π
sec–1x + cosec–1x = : x ∈R–[–1,1]
2
x+ y
6. tan–1x + tan–1y = tan–1 1 – xy : xy < 1
x− y
; xy > –1
tan–1x – tan–1y = tan–1 1 + xy
2x
7. 2tan–1x = sin–1 : –1 ≤ x ≤ 1
1 + x2
1 – x2
2tan–1x = cos–1 : x≥0
1 + x2
2x
2tan–1x = tan–1 : –1 < x < 1
1 – x2
2.2 Solved Examples
Short Answer (S.A.)
3
Example 1 Find the principal value of cos–1x, for x = .
2
