Inverse Trigonometric
Functions
Inverse Function
If y = f ( x ) and x = g ( y ) are two functions such that f ( g ( y )) = y and
g ( f ( y )) = x, then f and y are said to be inverse of each other,
i.e. g = f −1. If y = f ( x ), then x = f −1( y ).
Inverse Trigonometric Functions
As we know that trigonometric functions are not one-one and onto in
their natural domain and range, so their inverse do not exist but if we
restrict their domain and range, then their inverse may exists.
Domain and Range of Inverse Trigonometric Functions
The range of trigonometric functions becomes the domain of inverse
trigonometric functions and restricted domain of trigonometric
functions becomes range or principal value branch of inverse
trigonometric functions.
Table for Domain, Range and Other Possible
Range of Inverse Trigonometric Functions
Principal value
Function Domain Other possible range
branch (Range)
− π , π −3π , − π , π , 3π etc.
y = sin −1 x [ −1, 1] 2 2 2 2 2 2
y = cos −1 x [ −1, 1] [ 0, π ] [ −π , 0], [ π , 2π ] etc.
− π , π −3π , − π , π , 3π etc.
y = tan −1 x R
2 2 2 2 2 2
π π 3π
y = sec −1 x R − ( −1, 1) [ 0, π ] − [ −π , 0] − – , [ π , 2π ] − etc.
2 2 2
− π , π − {0} −3π , − π − {– π }, π , 3π − {π }
y = cosec −1 x R − ( −1, 1) 2 2 2 2 2 2
y = cot −1 x R ( 0, π ) ( −π , 0), ( π , 2π ) etc.
, Graphs of Inverse Trigonometric Functions
The graphs of inverse trigonometric functions with respect to line y = x
are given in the following table
Graph Graph
Function
(By interchanging axes) (By mirror image)
Y
5π
—–
2
2π Y
3π y=sin–1x
—– π y=x
2 —
π 2
1
π π y=sin x
— – — –1
2 1 2
−1
y = sin x X′ –1 X X′ π X
0π 0 1 —
–— 2
2 –1
–π π
3π –—
– —– 2
2
– 2π Y′
5π
– —–
2
Y′
Y
5π
—–
2
2π
3π Y
—– y=cos–1 x
2
π π
π/2 y=x
π/2
−1 1
y = cos x X′ X
1
–1 0 π
–—
2
y=cos x
–π X′ π X
–1 0 1 π/2
3π
– —–
2
–2π Y′
5π
– —–
2
Y′
Y 2π Y y=tan x
π
3— y=x
2 π/2
π y=tan–1 x
π –π/2
y = tan −1 x — X′ X
–2 –1 2 0 π/2
X′ –1 0 1 2 X
π –π/2
–— 2
–π
Y′ Y′
Functions
Inverse Function
If y = f ( x ) and x = g ( y ) are two functions such that f ( g ( y )) = y and
g ( f ( y )) = x, then f and y are said to be inverse of each other,
i.e. g = f −1. If y = f ( x ), then x = f −1( y ).
Inverse Trigonometric Functions
As we know that trigonometric functions are not one-one and onto in
their natural domain and range, so their inverse do not exist but if we
restrict their domain and range, then their inverse may exists.
Domain and Range of Inverse Trigonometric Functions
The range of trigonometric functions becomes the domain of inverse
trigonometric functions and restricted domain of trigonometric
functions becomes range or principal value branch of inverse
trigonometric functions.
Table for Domain, Range and Other Possible
Range of Inverse Trigonometric Functions
Principal value
Function Domain Other possible range
branch (Range)
− π , π −3π , − π , π , 3π etc.
y = sin −1 x [ −1, 1] 2 2 2 2 2 2
y = cos −1 x [ −1, 1] [ 0, π ] [ −π , 0], [ π , 2π ] etc.
− π , π −3π , − π , π , 3π etc.
y = tan −1 x R
2 2 2 2 2 2
π π 3π
y = sec −1 x R − ( −1, 1) [ 0, π ] − [ −π , 0] − – , [ π , 2π ] − etc.
2 2 2
− π , π − {0} −3π , − π − {– π }, π , 3π − {π }
y = cosec −1 x R − ( −1, 1) 2 2 2 2 2 2
y = cot −1 x R ( 0, π ) ( −π , 0), ( π , 2π ) etc.
, Graphs of Inverse Trigonometric Functions
The graphs of inverse trigonometric functions with respect to line y = x
are given in the following table
Graph Graph
Function
(By interchanging axes) (By mirror image)
Y
5π
—–
2
2π Y
3π y=sin–1x
—– π y=x
2 —
π 2
1
π π y=sin x
— – — –1
2 1 2
−1
y = sin x X′ –1 X X′ π X
0π 0 1 —
–— 2
2 –1
–π π
3π –—
– —– 2
2
– 2π Y′
5π
– —–
2
Y′
Y
5π
—–
2
2π
3π Y
—– y=cos–1 x
2
π π
π/2 y=x
π/2
−1 1
y = cos x X′ X
1
–1 0 π
–—
2
y=cos x
–π X′ π X
–1 0 1 π/2
3π
– —–
2
–2π Y′
5π
– —–
2
Y′
Y 2π Y y=tan x
π
3— y=x
2 π/2
π y=tan–1 x
π –π/2
y = tan −1 x — X′ X
–2 –1 2 0 π/2
X′ –1 0 1 2 X
π –π/2
–— 2
–π
Y′ Y′
