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1 | 818-821 | The dark portion of the graph of
y = sin–1 x represent the principal value branch (ii)
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 |
1 | 819-822 | (ii)
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 This can be visualised by looking the graphs of y = sin x and
y = sin–1 x as given in the same axes (Fig 2 |
1 | 820-823 | e , reflection) along
the line y = x This can be visualised by looking the graphs of y = sin x and
y = sin–1 x as given in the same axes (Fig 2 1 (iii)) |
1 | 821-824 | , reflection) along
the line y = x This can be visualised by looking the graphs of y = sin x and
y = sin–1 x as given in the same axes (Fig 2 1 (iii)) Like sine function, the cosine function is a function whose domain is the set of all
real numbers and range is the set [–1, 1] |
1 | 822-825 | This can be visualised by looking the graphs of y = sin x and
y = sin–1 x as given in the same axes (Fig 2 1 (iii)) Like sine function, the cosine function is a function whose domain is the set of all
real numbers and range is the set [–1, 1] If we restrict the domain of cosine function
to [0, π], then it becomes one-one and onto with range [–1, 1] |
1 | 823-826 | 1 (iii)) Like sine function, the cosine function is a function whose domain is the set of all
real numbers and range is the set [–1, 1] If we restrict the domain of cosine function
to [0, π], then it becomes one-one and onto with range [–1, 1] Actually, cosine function
Fig 2 |
1 | 824-827 | Like sine function, the cosine function is a function whose domain is the set of all
real numbers and range is the set [–1, 1] If we restrict the domain of cosine function
to [0, π], then it becomes one-one and onto with range [–1, 1] Actually, cosine function
Fig 2 1 (ii)
Fig 2 |
1 | 825-828 | If we restrict the domain of cosine function
to [0, π], then it becomes one-one and onto with range [–1, 1] Actually, cosine function
Fig 2 1 (ii)
Fig 2 1 (iii)
Fig 2 |
1 | 826-829 | Actually, cosine function
Fig 2 1 (ii)
Fig 2 1 (iii)
Fig 2 1 (i)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 21
restricted to any of the intervals [– π, 0], [0,π], [π, 2π] etc |
1 | 827-830 | 1 (ii)
Fig 2 1 (iii)
Fig 2 1 (i)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 21
restricted to any of the intervals [– π, 0], [0,π], [π, 2π] etc , is bijective with range as
[–1, 1] |
1 | 828-831 | 1 (iii)
Fig 2 1 (i)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 21
restricted to any of the intervals [– π, 0], [0,π], [π, 2π] etc , is bijective with range as
[–1, 1] We can, therefore, define the inverse of cosine function in each of these
intervals |
1 | 829-832 | 1 (i)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 21
restricted to any of the intervals [– π, 0], [0,π], [π, 2π] etc , is bijective with range as
[–1, 1] We can, therefore, define the inverse of cosine function in each of these
intervals We denote the inverse of the cosine function by cos–1 (arc cosine function) |
1 | 830-833 | , is bijective with range as
[–1, 1] We can, therefore, define the inverse of cosine function in each of these
intervals We denote the inverse of the cosine function by cos–1 (arc cosine function) Thus, cos–1 is a function whose domain is [–1, 1] and range
could be any of the intervals [–π, 0], [0, π], [π, 2π] etc |
1 | 831-834 | We can, therefore, define the inverse of cosine function in each of these
intervals We denote the inverse of the cosine function by cos–1 (arc cosine function) Thus, cos–1 is a function whose domain is [–1, 1] and range
could be any of the intervals [–π, 0], [0, π], [π, 2π] etc Corresponding to each such interval, we get a branch of the
function cos–1 |
1 | 832-835 | We denote the inverse of the cosine function by cos–1 (arc cosine function) Thus, cos–1 is a function whose domain is [–1, 1] and range
could be any of the intervals [–π, 0], [0, π], [π, 2π] etc Corresponding to each such interval, we get a branch of the
function cos–1 The branch with range [0, π] is called the principal
value branch of the function cos–1 |
1 | 833-836 | Thus, cos–1 is a function whose domain is [–1, 1] and range
could be any of the intervals [–π, 0], [0, π], [π, 2π] etc Corresponding to each such interval, we get a branch of the
function cos–1 The branch with range [0, π] is called the principal
value branch of the function cos–1 We write
cos–1 : [–1, 1] → [0, π] |
1 | 834-837 | Corresponding to each such interval, we get a branch of the
function cos–1 The branch with range [0, π] is called the principal
value branch of the function cos–1 We write
cos–1 : [–1, 1] → [0, π] The graph of the function given by y = cos–1 x can be drawn
in the same way as discussed about the graph of y = sin–1 x |
1 | 835-838 | The branch with range [0, π] is called the principal
value branch of the function cos–1 We write
cos–1 : [–1, 1] → [0, π] The graph of the function given by y = cos–1 x can be drawn
in the same way as discussed about the graph of y = sin–1 x The
graphs of y = cos x and y = cos–1x are given in Fig 2 |
1 | 836-839 | We write
cos–1 : [–1, 1] → [0, π] The graph of the function given by y = cos–1 x can be drawn
in the same way as discussed about the graph of y = sin–1 x The
graphs of y = cos x and y = cos–1x are given in Fig 2 2 (i) and (ii) |
1 | 837-840 | The graph of the function given by y = cos–1 x can be drawn
in the same way as discussed about the graph of y = sin–1 x The
graphs of y = cos x and y = cos–1x are given in Fig 2 2 (i) and (ii) Fig 2 |
1 | 838-841 | The
graphs of y = cos x and y = cos–1x are given in Fig 2 2 (i) and (ii) Fig 2 2 (ii)
Let us now discuss cosec–1x and sec–1x as follows:
Since, cosec x =
1
sin x , the domain of the cosec function is the set {x : x ∈ R and
x ≠ nπ, n ∈ Z} and the range is the set {y : y ∈ R, y ≥ 1 or y ≤ –1} i |
1 | 839-842 | 2 (i) and (ii) Fig 2 2 (ii)
Let us now discuss cosec–1x and sec–1x as follows:
Since, cosec x =
1
sin x , the domain of the cosec function is the set {x : x ∈ R and
x ≠ nπ, n ∈ Z} and the range is the set {y : y ∈ R, y ≥ 1 or y ≤ –1} i e |
1 | 840-843 | Fig 2 2 (ii)
Let us now discuss cosec–1x and sec–1x as follows:
Since, cosec x =
1
sin x , the domain of the cosec function is the set {x : x ∈ R and
x ≠ nπ, n ∈ Z} and the range is the set {y : y ∈ R, y ≥ 1 or y ≤ –1} i e , the set
R – (–1, 1) |
1 | 841-844 | 2 (ii)
Let us now discuss cosec–1x and sec–1x as follows:
Since, cosec x =
1
sin x , the domain of the cosec function is the set {x : x ∈ R and
x ≠ nπ, n ∈ Z} and the range is the set {y : y ∈ R, y ≥ 1 or y ≤ –1} i e , the set
R – (–1, 1) It means that y = cosec x assumes all real values except –1 < y < 1 and is
not defined for integral multiple of π |
1 | 842-845 | e , the set
R – (–1, 1) It means that y = cosec x assumes all real values except –1 < y < 1 and is
not defined for integral multiple of π If we restrict the domain of cosec function to
,
2 2
π π
−
– {0}, then it is one to one and onto with its range as the set R – (– 1, 1) |
1 | 843-846 | , the set
R – (–1, 1) It means that y = cosec x assumes all real values except –1 < y < 1 and is
not defined for integral multiple of π If we restrict the domain of cosec function to
,
2 2
π π
−
– {0}, then it is one to one and onto with its range as the set R – (– 1, 1) Actually,
cosec function restricted to any of the intervals
3 ,
{
}
2
2
− π −π
− −π
,
2,
2
−π π
– {0},
,3
{ }
2
2
π
π
− π
etc |
1 | 844-847 | It means that y = cosec x assumes all real values except –1 < y < 1 and is
not defined for integral multiple of π If we restrict the domain of cosec function to
,
2 2
π π
−
– {0}, then it is one to one and onto with its range as the set R – (– 1, 1) Actually,
cosec function restricted to any of the intervals
3 ,
{
}
2
2
− π −π
− −π
,
2,
2
−π π
– {0},
,3
{ }
2
2
π
π
− π
etc , is bijective and its range is the set of all real numbers R – (–1, 1) |
1 | 845-848 | If we restrict the domain of cosec function to
,
2 2
π π
−
– {0}, then it is one to one and onto with its range as the set R – (– 1, 1) Actually,
cosec function restricted to any of the intervals
3 ,
{
}
2
2
− π −π
− −π
,
2,
2
−π π
– {0},
,3
{ }
2
2
π
π
− π
etc , is bijective and its range is the set of all real numbers R – (–1, 1) Fig 2 |
1 | 846-849 | Actually,
cosec function restricted to any of the intervals
3 ,
{
}
2
2
− π −π
− −π
,
2,
2
−π π
– {0},
,3
{ }
2
2
π
π
− π
etc , is bijective and its range is the set of all real numbers R – (–1, 1) Fig 2 2 (i)
Rationalised 2023-24
22
MATHEMATICS
Thus cosec–1 can be defined as a function whose domain is R – (–1, 1) and range could
be any of the intervals
−
−
− −
23
2
π
π
π
,
{
},
−
2π π −
2
0
,
{ },
,3
{ }
2
2
π
π
− π
etc |
1 | 847-850 | , is bijective and its range is the set of all real numbers R – (–1, 1) Fig 2 2 (i)
Rationalised 2023-24
22
MATHEMATICS
Thus cosec–1 can be defined as a function whose domain is R – (–1, 1) and range could
be any of the intervals
−
−
− −
23
2
π
π
π
,
{
},
−
2π π −
2
0
,
{ },
,3
{ }
2
2
π
π
− π
etc The
function corresponding to the range
,
{0}
2
−π π2
−
is called the principal value branch
of cosec–1 |
1 | 848-851 | Fig 2 2 (i)
Rationalised 2023-24
22
MATHEMATICS
Thus cosec–1 can be defined as a function whose domain is R – (–1, 1) and range could
be any of the intervals
−
−
− −
23
2
π
π
π
,
{
},
−
2π π −
2
0
,
{ },
,3
{ }
2
2
π
π
− π
etc The
function corresponding to the range
,
{0}
2
−π π2
−
is called the principal value branch
of cosec–1 We thus have principal branch as
cosec–1 : R – (–1, 1) →
,
{0}
2
−π π2
−
The graphs of y = cosec x and y = cosec–1 x are given in Fig 2 |
1 | 849-852 | 2 (i)
Rationalised 2023-24
22
MATHEMATICS
Thus cosec–1 can be defined as a function whose domain is R – (–1, 1) and range could
be any of the intervals
−
−
− −
23
2
π
π
π
,
{
},
−
2π π −
2
0
,
{ },
,3
{ }
2
2
π
π
− π
etc The
function corresponding to the range
,
{0}
2
−π π2
−
is called the principal value branch
of cosec–1 We thus have principal branch as
cosec–1 : R – (–1, 1) →
,
{0}
2
−π π2
−
The graphs of y = cosec x and y = cosec–1 x are given in Fig 2 3 (i), (ii) |
1 | 850-853 | The
function corresponding to the range
,
{0}
2
−π π2
−
is called the principal value branch
of cosec–1 We thus have principal branch as
cosec–1 : R – (–1, 1) →
,
{0}
2
−π π2
−
The graphs of y = cosec x and y = cosec–1 x are given in Fig 2 3 (i), (ii) Also, since sec x =
1
cos x , the domain of y = sec x is the set R – {x : x = (2n + 1) 2
π ,
n ∈ Z} and range is the set R – (–1, 1) |
1 | 851-854 | We thus have principal branch as
cosec–1 : R – (–1, 1) →
,
{0}
2
−π π2
−
The graphs of y = cosec x and y = cosec–1 x are given in Fig 2 3 (i), (ii) Also, since sec x =
1
cos x , the domain of y = sec x is the set R – {x : x = (2n + 1) 2
π ,
n ∈ Z} and range is the set R – (–1, 1) It means that sec (secant function) assumes
all real values except –1 < y < 1 and is not defined for odd multiples of 2
π |
1 | 852-855 | 3 (i), (ii) Also, since sec x =
1
cos x , the domain of y = sec x is the set R – {x : x = (2n + 1) 2
π ,
n ∈ Z} and range is the set R – (–1, 1) It means that sec (secant function) assumes
all real values except –1 < y < 1 and is not defined for odd multiples of 2
π If we
restrict the domain of secant function to [0, π] – { 2
π }, then it is one-one and onto with
Fig 2 |
1 | 853-856 | Also, since sec x =
1
cos x , the domain of y = sec x is the set R – {x : x = (2n + 1) 2
π ,
n ∈ Z} and range is the set R – (–1, 1) It means that sec (secant function) assumes
all real values except –1 < y < 1 and is not defined for odd multiples of 2
π If we
restrict the domain of secant function to [0, π] – { 2
π }, then it is one-one and onto with
Fig 2 3 (i)
Fig 2 |
1 | 854-857 | It means that sec (secant function) assumes
all real values except –1 < y < 1 and is not defined for odd multiples of 2
π If we
restrict the domain of secant function to [0, π] – { 2
π }, then it is one-one and onto with
Fig 2 3 (i)
Fig 2 3 (ii)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 23
its range as the set R – (–1, 1) |
1 | 855-858 | If we
restrict the domain of secant function to [0, π] – { 2
π }, then it is one-one and onto with
Fig 2 3 (i)
Fig 2 3 (ii)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 23
its range as the set R – (–1, 1) Actually, secant function restricted to any of the
intervals [–π, 0] – { 2
−π }, [0, ] –
π2
π
, [π, 2π] – { 3
2
π } etc |
1 | 856-859 | 3 (i)
Fig 2 3 (ii)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 23
its range as the set R – (–1, 1) Actually, secant function restricted to any of the
intervals [–π, 0] – { 2
−π }, [0, ] –
π2
π
, [π, 2π] – { 3
2
π } etc , is bijective and its range
is R – {–1, 1} |
1 | 857-860 | 3 (ii)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 23
its range as the set R – (–1, 1) Actually, secant function restricted to any of the
intervals [–π, 0] – { 2
−π }, [0, ] –
π2
π
, [π, 2π] – { 3
2
π } etc , is bijective and its range
is R – {–1, 1} Thus sec–1 can be defined as a function whose domain is R– (–1, 1) and
range could be any of the intervals [– π, 0] – { 2
−π }, [0, π] – { 2
π }, [π, 2π] – { 3
2
π } etc |
1 | 858-861 | Actually, secant function restricted to any of the
intervals [–π, 0] – { 2
−π }, [0, ] –
π2
π
, [π, 2π] – { 3
2
π } etc , is bijective and its range
is R – {–1, 1} Thus sec–1 can be defined as a function whose domain is R– (–1, 1) and
range could be any of the intervals [– π, 0] – { 2
−π }, [0, π] – { 2
π }, [π, 2π] – { 3
2
π } etc Corresponding to each of these intervals, we get different branches of the function sec–1 |
1 | 859-862 | , is bijective and its range
is R – {–1, 1} Thus sec–1 can be defined as a function whose domain is R– (–1, 1) and
range could be any of the intervals [– π, 0] – { 2
−π }, [0, π] – { 2
π }, [π, 2π] – { 3
2
π } etc Corresponding to each of these intervals, we get different branches of the function sec–1 The branch with range [0, π] – { 2
π } is called the principal value branch of the
function sec–1 |
1 | 860-863 | Thus sec–1 can be defined as a function whose domain is R– (–1, 1) and
range could be any of the intervals [– π, 0] – { 2
−π }, [0, π] – { 2
π }, [π, 2π] – { 3
2
π } etc Corresponding to each of these intervals, we get different branches of the function sec–1 The branch with range [0, π] – { 2
π } is called the principal value branch of the
function sec–1 We thus have
sec–1 : R – (–1,1) → [0, π] – { 2
π }
The graphs of the functions y = sec x and y = sec-1 x are given in Fig 2 |
1 | 861-864 | Corresponding to each of these intervals, we get different branches of the function sec–1 The branch with range [0, π] – { 2
π } is called the principal value branch of the
function sec–1 We thus have
sec–1 : R – (–1,1) → [0, π] – { 2
π }
The graphs of the functions y = sec x and y = sec-1 x are given in Fig 2 4 (i), (ii) |
1 | 862-865 | The branch with range [0, π] – { 2
π } is called the principal value branch of the
function sec–1 We thus have
sec–1 : R – (–1,1) → [0, π] – { 2
π }
The graphs of the functions y = sec x and y = sec-1 x are given in Fig 2 4 (i), (ii) Finally, we now discuss tan–1 and cot–1
We know that the domain of the tan function (tangent function) is the set
{x : x ∈ R and x ≠ (2n +1) 2
π , n ∈ Z} and the range is R |
1 | 863-866 | We thus have
sec–1 : R – (–1,1) → [0, π] – { 2
π }
The graphs of the functions y = sec x and y = sec-1 x are given in Fig 2 4 (i), (ii) Finally, we now discuss tan–1 and cot–1
We know that the domain of the tan function (tangent function) is the set
{x : x ∈ R and x ≠ (2n +1) 2
π , n ∈ Z} and the range is R It means that tan function
is not defined for odd multiples of 2
π |
1 | 864-867 | 4 (i), (ii) Finally, we now discuss tan–1 and cot–1
We know that the domain of the tan function (tangent function) is the set
{x : x ∈ R and x ≠ (2n +1) 2
π , n ∈ Z} and the range is R It means that tan function
is not defined for odd multiples of 2
π If we restrict the domain of tangent function to
Fig 2 |
1 | 865-868 | Finally, we now discuss tan–1 and cot–1
We know that the domain of the tan function (tangent function) is the set
{x : x ∈ R and x ≠ (2n +1) 2
π , n ∈ Z} and the range is R It means that tan function
is not defined for odd multiples of 2
π If we restrict the domain of tangent function to
Fig 2 4 (i)
Fig 2 |
1 | 866-869 | It means that tan function
is not defined for odd multiples of 2
π If we restrict the domain of tangent function to
Fig 2 4 (i)
Fig 2 4 (ii)
Rationalised 2023-24
24
MATHEMATICS
2,
2
−π π
, then it is one-one and onto with its range as R |
1 | 867-870 | If we restrict the domain of tangent function to
Fig 2 4 (i)
Fig 2 4 (ii)
Rationalised 2023-24
24
MATHEMATICS
2,
2
−π π
, then it is one-one and onto with its range as R Actually, tangent function
restricted to any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
etc |
1 | 868-871 | 4 (i)
Fig 2 4 (ii)
Rationalised 2023-24
24
MATHEMATICS
2,
2
−π π
, then it is one-one and onto with its range as R Actually, tangent function
restricted to any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
etc , is bijective
and its range is R |
1 | 869-872 | 4 (ii)
Rationalised 2023-24
24
MATHEMATICS
2,
2
−π π
, then it is one-one and onto with its range as R Actually, tangent function
restricted to any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
etc , is bijective
and its range is R Thus tan–1 can be defined as a function whose domain is R and
range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
and so on |
1 | 870-873 | Actually, tangent function
restricted to any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
etc , is bijective
and its range is R Thus tan–1 can be defined as a function whose domain is R and
range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
and so on These
intervals give different branches of the function tan–1 |
1 | 871-874 | , is bijective
and its range is R Thus tan–1 can be defined as a function whose domain is R and
range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
and so on These
intervals give different branches of the function tan–1 The branch with range
2,
2
−π π
is called the principal value branch of the function tan–1 |
1 | 872-875 | Thus tan–1 can be defined as a function whose domain is R and
range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
,
2,3
2
π
π
and so on These
intervals give different branches of the function tan–1 The branch with range
2,
2
−π π
is called the principal value branch of the function tan–1 We thus have
tan–1 : R →
2,
2
−π π
The graphs of the function y = tan x and y = tan–1x are given in Fig 2 |
1 | 873-876 | These
intervals give different branches of the function tan–1 The branch with range
2,
2
−π π
is called the principal value branch of the function tan–1 We thus have
tan–1 : R →
2,
2
−π π
The graphs of the function y = tan x and y = tan–1x are given in Fig 2 5 (i), (ii) |
1 | 874-877 | The branch with range
2,
2
−π π
is called the principal value branch of the function tan–1 We thus have
tan–1 : R →
2,
2
−π π
The graphs of the function y = tan x and y = tan–1x are given in Fig 2 5 (i), (ii) Fig 2 |
1 | 875-878 | We thus have
tan–1 : R →
2,
2
−π π
The graphs of the function y = tan x and y = tan–1x are given in Fig 2 5 (i), (ii) Fig 2 5 (i)
Fig 2 |
1 | 876-879 | 5 (i), (ii) Fig 2 5 (i)
Fig 2 5 (ii)
We know that domain of the cot function (cotangent function) is the set
{x : x ∈ R and x ≠ nπ, n ∈ Z} and range is R |
1 | 877-880 | Fig 2 5 (i)
Fig 2 5 (ii)
We know that domain of the cot function (cotangent function) is the set
{x : x ∈ R and x ≠ nπ, n ∈ Z} and range is R It means that cotangent function is not
defined for integral multiples of π |
1 | 878-881 | 5 (i)
Fig 2 5 (ii)
We know that domain of the cot function (cotangent function) is the set
{x : x ∈ R and x ≠ nπ, n ∈ Z} and range is R It means that cotangent function is not
defined for integral multiples of π If we restrict the domain of cotangent function to
(0, π), then it is bijective with and its range as R |
1 | 879-882 | 5 (ii)
We know that domain of the cot function (cotangent function) is the set
{x : x ∈ R and x ≠ nπ, n ∈ Z} and range is R It means that cotangent function is not
defined for integral multiples of π If we restrict the domain of cotangent function to
(0, π), then it is bijective with and its range as R In fact, cotangent function restricted
to any of the intervals (–π, 0), (0, π), (π, 2π) etc |
1 | 880-883 | It means that cotangent function is not
defined for integral multiples of π If we restrict the domain of cotangent function to
(0, π), then it is bijective with and its range as R In fact, cotangent function restricted
to any of the intervals (–π, 0), (0, π), (π, 2π) etc , is bijective and its range is R |
1 | 881-884 | If we restrict the domain of cotangent function to
(0, π), then it is bijective with and its range as R In fact, cotangent function restricted
to any of the intervals (–π, 0), (0, π), (π, 2π) etc , is bijective and its range is R Thus
cot –1 can be defined as a function whose domain is the R and range as any of the
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 25
intervals (–π, 0), (0, π), (π, 2π) etc |
1 | 882-885 | In fact, cotangent function restricted
to any of the intervals (–π, 0), (0, π), (π, 2π) etc , is bijective and its range is R Thus
cot –1 can be defined as a function whose domain is the R and range as any of the
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 25
intervals (–π, 0), (0, π), (π, 2π) etc These intervals give different branches of the
function cot –1 |
1 | 883-886 | , is bijective and its range is R Thus
cot –1 can be defined as a function whose domain is the R and range as any of the
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 25
intervals (–π, 0), (0, π), (π, 2π) etc These intervals give different branches of the
function cot –1 The function with range (0, π) is called the principal value branch of
the function cot –1 |
1 | 884-887 | Thus
cot –1 can be defined as a function whose domain is the R and range as any of the
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 25
intervals (–π, 0), (0, π), (π, 2π) etc These intervals give different branches of the
function cot –1 The function with range (0, π) is called the principal value branch of
the function cot –1 We thus have
cot–1 : R → (0, π)
The graphs of y = cot x and y = cot–1x are given in Fig 2 |
1 | 885-888 | These intervals give different branches of the
function cot –1 The function with range (0, π) is called the principal value branch of
the function cot –1 We thus have
cot–1 : R → (0, π)
The graphs of y = cot x and y = cot–1x are given in Fig 2 6 (i), (ii) |
1 | 886-889 | The function with range (0, π) is called the principal value branch of
the function cot –1 We thus have
cot–1 : R → (0, π)
The graphs of y = cot x and y = cot–1x are given in Fig 2 6 (i), (ii) Fig 2 |
1 | 887-890 | We thus have
cot–1 : R → (0, π)
The graphs of y = cot x and y = cot–1x are given in Fig 2 6 (i), (ii) Fig 2 6 (i)
Fig 2 |
1 | 888-891 | 6 (i), (ii) Fig 2 6 (i)
Fig 2 6 (ii)
The following table gives the inverse trigonometric function (principal value
branches) along with their domains and ranges |
1 | 889-892 | Fig 2 6 (i)
Fig 2 6 (ii)
The following table gives the inverse trigonometric function (principal value
branches) along with their domains and ranges sin–1
:
[–1, 1]
→
,
2 2
π π
−
cos–1
:
[–1, 1]
→
[0, π]
cosec–1
:
R – (–1,1)
→
,
2 2
π π
−
– {0}
sec–1
:
R – (–1, 1)
→
[0, π] – { }
2
π
tan–1
:
R
→
2,
2
−π π
cot–1
:
R
→
(0, π)
Rationalised 2023-24
26
MATHEMATICS
ANote
1 |
1 | 890-893 | 6 (i)
Fig 2 6 (ii)
The following table gives the inverse trigonometric function (principal value
branches) along with their domains and ranges sin–1
:
[–1, 1]
→
,
2 2
π π
−
cos–1
:
[–1, 1]
→
[0, π]
cosec–1
:
R – (–1,1)
→
,
2 2
π π
−
– {0}
sec–1
:
R – (–1, 1)
→
[0, π] – { }
2
π
tan–1
:
R
→
2,
2
−π π
cot–1
:
R
→
(0, π)
Rationalised 2023-24
26
MATHEMATICS
ANote
1 sin–1x should not be confused with (sin x)–1 |
1 | 891-894 | 6 (ii)
The following table gives the inverse trigonometric function (principal value
branches) along with their domains and ranges sin–1
:
[–1, 1]
→
,
2 2
π π
−
cos–1
:
[–1, 1]
→
[0, π]
cosec–1
:
R – (–1,1)
→
,
2 2
π π
−
– {0}
sec–1
:
R – (–1, 1)
→
[0, π] – { }
2
π
tan–1
:
R
→
2,
2
−π π
cot–1
:
R
→
(0, π)
Rationalised 2023-24
26
MATHEMATICS
ANote
1 sin–1x should not be confused with (sin x)–1 In fact (sin x)–1 =
1
sin x and
similarly for other trigonometric functions |
1 | 892-895 | sin–1
:
[–1, 1]
→
,
2 2
π π
−
cos–1
:
[–1, 1]
→
[0, π]
cosec–1
:
R – (–1,1)
→
,
2 2
π π
−
– {0}
sec–1
:
R – (–1, 1)
→
[0, π] – { }
2
π
tan–1
:
R
→
2,
2
−π π
cot–1
:
R
→
(0, π)
Rationalised 2023-24
26
MATHEMATICS
ANote
1 sin–1x should not be confused with (sin x)–1 In fact (sin x)–1 =
1
sin x and
similarly for other trigonometric functions 2 |
1 | 893-896 | sin–1x should not be confused with (sin x)–1 In fact (sin x)–1 =
1
sin x and
similarly for other trigonometric functions 2 Whenever no branch of an inverse trigonometric functions is mentioned, we
mean the principal value branch of that function |
1 | 894-897 | In fact (sin x)–1 =
1
sin x and
similarly for other trigonometric functions 2 Whenever no branch of an inverse trigonometric functions is mentioned, we
mean the principal value branch of that function 3 |
1 | 895-898 | 2 Whenever no branch of an inverse trigonometric functions is mentioned, we
mean the principal value branch of that function 3 The value of an inverse trigonometric functions which lies in the range of
principal branch is called the principal value of that inverse trigonometric
functions |
1 | 896-899 | Whenever no branch of an inverse trigonometric functions is mentioned, we
mean the principal value branch of that function 3 The value of an inverse trigonometric functions which lies in the range of
principal branch is called the principal value of that inverse trigonometric
functions We now consider some examples:
Example 1 Find the principal value of sin–1
1
2
|
1 | 897-900 | 3 The value of an inverse trigonometric functions which lies in the range of
principal branch is called the principal value of that inverse trigonometric
functions We now consider some examples:
Example 1 Find the principal value of sin–1
1
2
Solution Let sin–1
1
2
= y |
1 | 898-901 | The value of an inverse trigonometric functions which lies in the range of
principal branch is called the principal value of that inverse trigonometric
functions We now consider some examples:
Example 1 Find the principal value of sin–1
1
2
Solution Let sin–1
1
2
= y Then, sin y = 1
2 |
1 | 899-902 | We now consider some examples:
Example 1 Find the principal value of sin–1
1
2
Solution Let sin–1
1
2
= y Then, sin y = 1
2 We know that the range of the principal value branch of sin–1 is
−
2π π
,2
and
sin 4
π
=
1
2 |
1 | 900-903 | Solution Let sin–1
1
2
= y Then, sin y = 1
2 We know that the range of the principal value branch of sin–1 is
−
2π π
,2
and
sin 4
π
=
1
2 Therefore, principal value of sin–1
1
2
is 4
π
Example 2 Find the principal value of cot–1
1
3
−
Solution Let cot–1
1
3
−
= y |
1 | 901-904 | Then, sin y = 1
2 We know that the range of the principal value branch of sin–1 is
−
2π π
,2
and
sin 4
π
=
1
2 Therefore, principal value of sin–1
1
2
is 4
π
Example 2 Find the principal value of cot–1
1
3
−
Solution Let cot–1
1
3
−
= y Then,
1
cot
cot 3
3
y
−
π
=
= −
= cot
π3
π −
=
cot2
π3
We know that the range of principal value branch of cot–1 is (0, π) and
cot 2
π3
=
1
3
− |
1 | 902-905 | We know that the range of the principal value branch of sin–1 is
−
2π π
,2
and
sin 4
π
=
1
2 Therefore, principal value of sin–1
1
2
is 4
π
Example 2 Find the principal value of cot–1
1
3
−
Solution Let cot–1
1
3
−
= y Then,
1
cot
cot 3
3
y
−
π
=
= −
= cot
π3
π −
=
cot2
π3
We know that the range of principal value branch of cot–1 is (0, π) and
cot 2
π3
=
1
3
− Hence, principal value of cot–1
1
3
−
is 2
3
π
EXERCISE 2 |
1 | 903-906 | Therefore, principal value of sin–1
1
2
is 4
π
Example 2 Find the principal value of cot–1
1
3
−
Solution Let cot–1
1
3
−
= y Then,
1
cot
cot 3
3
y
−
π
=
= −
= cot
π3
π −
=
cot2
π3
We know that the range of principal value branch of cot–1 is (0, π) and
cot 2
π3
=
1
3
− Hence, principal value of cot–1
1
3
−
is 2
3
π
EXERCISE 2 1
Find the principal values of the following:
1 |
1 | 904-907 | Then,
1
cot
cot 3
3
y
−
π
=
= −
= cot
π3
π −
=
cot2
π3
We know that the range of principal value branch of cot–1 is (0, π) and
cot 2
π3
=
1
3
− Hence, principal value of cot–1
1
3
−
is 2
3
π
EXERCISE 2 1
Find the principal values of the following:
1 sin–1
1
2
−
2 |
1 | 905-908 | Hence, principal value of cot–1
1
3
−
is 2
3
π
EXERCISE 2 1
Find the principal values of the following:
1 sin–1
1
2
−
2 cos–1
3
2
3 |
1 | 906-909 | 1
Find the principal values of the following:
1 sin–1
1
2
−
2 cos–1
3
2
3 cosec–1 (2)
4 |
1 | 907-910 | sin–1
1
2
−
2 cos–1
3
2
3 cosec–1 (2)
4 tan–1 (
−3)
5 |
1 | 908-911 | cos–1
3
2
3 cosec–1 (2)
4 tan–1 (
−3)
5 cos–1
1
2
−
6 |
1 | 909-912 | cosec–1 (2)
4 tan–1 (
−3)
5 cos–1
1
2
−
6 tan–1 (–1)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 27
7 |
1 | 910-913 | tan–1 (
−3)
5 cos–1
1
2
−
6 tan–1 (–1)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 27
7 sec–1
2
3
8 |
1 | 911-914 | cos–1
1
2
−
6 tan–1 (–1)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 27
7 sec–1
2
3
8 cot–1 ( 3)
9 |
1 | 912-915 | tan–1 (–1)
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 27
7 sec–1
2
3
8 cot–1 ( 3)
9 cos–1
1
2
−
10 |
1 | 913-916 | sec–1
2
3
8 cot–1 ( 3)
9 cos–1
1
2
−
10 cosec–1 (
−2
)
Find the values of the following:
11 |
1 | 914-917 | cot–1 ( 3)
9 cos–1
1
2
−
10 cosec–1 (
−2
)
Find the values of the following:
11 tan–1(1) + cos–1
1
2
−
+ sin–1
1
2
−
12 |
1 | 915-918 | cos–1
1
2
−
10 cosec–1 (
−2
)
Find the values of the following:
11 tan–1(1) + cos–1
1
2
−
+ sin–1
1
2
−
12 cos–1 1
2
+ 2 sin–1 1
2
13 |
1 | 916-919 | cosec–1 (
−2
)
Find the values of the following:
11 tan–1(1) + cos–1
1
2
−
+ sin–1
1
2
−
12 cos–1 1
2
+ 2 sin–1 1
2
13 If sin–1 x = y, then
(A) 0 ≤ y ≤ π
(B)
2
2
y
π
π
−
≤
≤
(C) 0 < y < π
(D)
2
2
y
π
π
−
<
<
14 |
1 | 917-920 | tan–1(1) + cos–1
1
2
−
+ sin–1
1
2
−
12 cos–1 1
2
+ 2 sin–1 1
2
13 If sin–1 x = y, then
(A) 0 ≤ y ≤ π
(B)
2
2
y
π
π
−
≤
≤
(C) 0 < y < π
(D)
2
2
y
π
π
−
<
<
14 tan–1
(
)
1
3
sec
2
−
−
−
is equal to
(A) π
(B)
−π3
(C)
π3
(D) 2
3
π
2 |
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