v
Mathematics, in general, is fundamentally the science of
self-evident things. — FELIX KLEIN
v
2.1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f
–1
, exists if f is one-one and onto. There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses. In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist. In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations.
Besides, some elementary properties will also be discussed.
The inverse trigonometric functions play an important
role in calculus for they serve to define many integrals.
The concepts of inverse trigonometric functions is also used in science and engineering.
2.2 Basic Concepts
In Class XI, we have studied trigonometric functions, which are defined as follows:
sine function, i.e., sine : R → [– 1, 1]
cosine function, i.e., cos : R → [– 1, 1]
tangent function, i.e., tan : R – { x : x = (2n + 1)
2
π
, n ∈ Z} → R
cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R
secant function, i.e., sec : R – { x : x = (2n + 1)
2
π
, n ∈ Z} → R – (– 1, 1)
cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)
Chapter
2
INVERSE TRIGONOMETRIC
FUNCTIONS
Aryabhata
(476-550 A. D.)
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34 MATHEMATICS
We have also learnt in Chapter 1 that if f : X→Y such that f(x) = y is one-one and
onto, then we can define a unique function g : Y→X such that g(y) = x, where
x ∈ X
and y = f(x), y ∈
Y. Here, the domain of g = range of f and the range of g = domain
of f. The function g is called the inverse of f and is denoted by f
–1
. Further, g is also
one-one and onto and inverse of g is f. Thus, g
–1
= (f
–1
)
–1
= f. We also have
(f
–1
o f ) (x) = f
–1
(f (x)) =
f
–1
(y) = x
and (f o f
–1
) (y) = f (f
–1
(y))
= f(x) = y
Since the domain of sine function is the set of all real numbers and range is the
closed interval [–1, 1]. If we restrict its domain to
,
2 2
−π π
, then it becomes one-one
and onto with range [– 1, 1]. Actually, sine function restricted to any of the intervals
−
3
2 2
π π
,
,
,
2 2
,
3
,
2 2
π π
etc., is one-one and its range is [–1, 1]. We can,
therefore, define the inverse of sine function in each of these intervals. We denote the
inverse of sine function by sin
–1
(arc sine function). Thus, sin
–1
is a function whose
domain is [– 1, 1] and range could be any of the intervals
3
,
2 2
− π −π
,
,
2 2
or
3
,
2 2
π π
, and so on. Corresponding to each such interval, we get a branch of the
function sin
–1
. The branch with range
,
2 2
−π π
is called the principal value branch,
whereas other intervals as range give different branches of sin
–1
. When we refer
to the function sin
–1
, we take it as the function whose domain is [–1, 1] and range is
,
2 2
−π π
. We write sin
–1
: [–1, 1] →
,
2 2
−π π
From the definition of the inverse functions, it follows that sin (sin
–1
x) = x
if – 1 ≤ x ≤ 1 and sin
–1
(sin x) = x if
2 2
x
π π
− ≤ ≤
. In other words, if y = sin
–1
x, then
sin y = x.
Remarks
(i) We know from Chapter 1, that if y = f(x) is an invertible function, then x = f
–1
(y).
Thus, the graph of sin
–1
function can be obtained from the graph of original
function by interchanging x and y axes, i.e., if (a, b) is a point on the graph of
sine function, then (b, a) becomes the corresponding point on the graph of inverse
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INVERSE TRIGONOMETRIC FUNCTIONS 35
of sine function. Thus, the graph of the function y = sin
–1
x can be obtained from
the graph of y = sin x by interchanging x and y axes. The graphs of y = sin x and
y = sin
–1
x are as given in Fig 2.1 (i), (ii), (iii). 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. 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]. Actually, cosine function
Fig 2.1 (ii)
Fig 2.1 (iii)
Fig 2.1 (i)
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36 MATHEMATICS
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).
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].
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
–1
x are given in Fig
2.2 (i) and (ii).
Fig 2.2 (ii)
Let us now discuss cosec
–1
x and sec
–1
x 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. 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 2.2 (i)
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INVERSE TRIGONOMETRIC FUNCTIONS 37
Thus cosec
–1
can be defined as a function whose domain is R – (–1, 1) and range could
be any of the intervals
− −
− −
3
2 2
π π
π, { }
,
−
−
π π
2 2
0, { }
,
3
, { }
2 2
π π
− π
etc. The
function corresponding to the range
, {0}
2 2
−π π
−
is called the principal value branch
of cosec
–1
. We thus have principal branch as
cosec
–1
: R – (–1, 1) →
, {0}
2 2
−π π
−
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
. 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)
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38 MATHEMATICS
its range as the set R – (–1, 1). Actually, secant function restricted to any of the
intervals [–π, 0] – {
2
−π
},
[0, ] –
2
π
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
.
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. 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)
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INVERSE TRIGONOMETRIC FUNCTIONS 39
,
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
3
,
2 2
− π −π
,
,
2 2
−π π
,
3
,
2 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
3
,
2 2
− π −π
,
,
2 2
−π π
,
3
,
2 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
–1
x are given in Fig 2.5 (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. 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. Thus
cot
–1
can be defined as a function whose domain is the R and range as any of the
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40 MATHEMATICS
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
–1
x are given in Fig 2.6 (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.
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)
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INVERSE TRIGONOMETRIC FUNCTIONS 41
A
Note
1. sin
–1
x 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.
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. Then, sin y =
1
2
.
We know that the range of the principal value branch of sin
–1
is
−
π π
2 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
3
π
π −
=
2
cot
3
π
We know that the range of principal value branch of cot
–1
is (0, π) and
cot
2
3
π
=
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. cos
–1
3
2
3. cosec
–1
(2)
4. tan
–1
( 3)
−
5. cos
–1
1
2
−
6. tan
–1
(–1)
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42 MATHEMATICS
7. sec
–1
2
3
8. 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. 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)
3
π
−
(C)
3
π
(D)
2
3
π
2.3 Properties of Inverse Trigonometric Functions
In this section, we shall prove some important properties of inverse trigonometric
functions. It may be mentioned here that these results are valid within the principal
value branches of the corresponding inverse trigonometric functions and wherever
they are defined. Some results may not be valid for all values of the domains of inverse
trigonometric functions. In fact, they will be valid only for some values of x for which
inverse trigonometric functions are defined. We will not go into the details of these
values of x in the domain as this discussion goes beyond the scope of this text book.
Let us recall that if y = sin
–1
x, then x = sin y and if x = sin y, then y = sin
–1
x. This is
equivalent to
sin (sin
–1
x) = x, x ∈ [– 1, 1] and sin
–1
(sin x) = x, x ∈
,
2 2
π π
−
Same is true for other five inverse trigonometric functions as well. We now prove
some properties of inverse trigonometric functions.
1. (i) sin
–1
1
x
= cosec
–1
x, x
≥≥
≥≥
≥ 1 or x
≤≤
≤≤
≤ – 1
(ii) cos
–1
1
x
= sec
–1
x, x
≥≥
≥≥
≥ 1 or x
≤≤
≤≤
≤ – 1
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INVERSE TRIGONOMETRIC FUNCTIONS 43
(iii) tan
–1
1
x
= cot
–1
x, x > 0
To prove the first result, we put cosec
–1
x = y, i.e., x = cosec y
Therefore
1
x
= sin y
Hence sin
–1
1
x
= y
or sin
–1
1
x
= cosec
–1
x
Similarly, we can prove the other parts.
2. (i) sin
–1
(–x) = – sin
–1
x, x
∈∈
∈∈
∈ [– 1, 1]
(ii) tan
–1
(–x) = – tan
–1
x, x
∈∈
∈∈
∈ R
(iii) cosec
–1
(–x) = – cosec
–1
x, |x|
≥≥
≥≥
≥ 1
Let sin
–1
(–x) = y, i.e., –x = sin y so that x = – sin y, i.e., x = sin (–y).
Hence sin
–1
x = – y = – sin
–1
(–x)
Therefore sin
–1
(–x) = – sin
–1
x
Similarly, we can prove the other parts.
3. (i) cos
–1
(–x) =
π
π
ππ
π – cos
–1
x, x
∈∈
∈∈
∈ [– 1, 1]
(ii) sec
–1
(–x) =
ππ
ππ
π – sec
–1
x, |x|
≥≥
≥≥
≥ 1
(iii) cot
–1
(–x) =
ππ
ππ
π – cot
–1
x, x
∈∈
∈∈
∈ R
Let cos
–1
(–x) = y i.e., – x = cos y so that x = – cos y = cos (π – y)
Therefore cos
–1
x = π – y = π – cos
–1
(–x)
Hence cos
–1
(–x) = π – cos
–1
x
Similarly, we can prove the other parts.
4. (i) sin
–1
x + cos
–1
x =
2
π
ππ
π
, x
∈∈
∈∈
∈ [– 1, 1]
(ii) tan
–1
x + cot
–1
x =
2
π
ππ
π
, x
∈∈
∈∈
∈ R
(iii) cosec
–1
x + sec
–1
x =
2
π
ππ
π
, |x|
≥≥
≥≥
≥ 1
Let sin
–1
x = y. Then x = sin y = cos
2
y
π
−
Therefore cos
–1
x =
2
y
π
−
=
–1
sin
2
x
π
−
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44 MATHEMATICS
Hence sin
–1
x + cos
–1
x =
2
π
Similarly, we can prove the other parts.
5. (i) tan
–1
x + tan
–1
y = tan
–1
1
x + y
– xy
, xy < 1
(ii) tan
–1
x – tan
–1
y = tan
–1
1
x – y
+ xy
, xy > – 1
(iii) tan
–1
x + tan
–1
y = π + tan
–1
, xy > 1; x, y > 0
Let tan
–1
x = θ and tan
–1
y = φ. Then x = tan θ, y = tan φ
Now
tan tan
tan( )
1 tan tan 1
x y
xy
θ+ φ +
θ+φ = =
− θ φ −
This gives θ + φ = tan
–1
1
x y
xy
+
−
Hence tan
–1
x + tan
–1
y = tan
–1
1
x y
xy
+
−
In the above result, if we replace y by – y, we get the second result and by replacing
y by x, we get the third result as given below.
6. (i) 2tan
–1
x = sin
–1
2
2
1 +
x
x
, |x|
≤≤
≤≤
≤ 1
(ii) 2tan
–1
x = cos
–1
2
2
1 –
1 +
x
x
, x
≥≥
≥≥
≥ 0
(iii) 2 tan
–1
x = tan
–1
2
2
1 –
x
x
, – 1 < x < 1
Let tan
–1
x = y, then x = tan y. Now
sin
–1
2
2
1
x
x
+
= sin
–1
2
2 tan
1 tan
y
y
+
= sin
–1
(sin 2y) = 2y = 2tan
–1
x
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INVERSE TRIGONOMETRIC FUNCTIONS 45
Also cos
–1
2
2
1
1
x
x
−
+
= cos
–1
2
2
1 tan
1 tan
y
y
−
+
= cos
–1
(cos 2y) = 2y = 2tan
–1
x
(iii) Can be worked out similarly.
We now consider some examples.
Example 3 Show that
(i) sin
–1
(
)
2
2 1
x x
−
= 2 sin
–1
x,
1 1
2 2
x− ≤ ≤
(ii) sin
–1
(
)
2
2 1
x x
−
= 2 cos
–1
x,
1
1
2
x
≤ ≤
Solution
(i) Let x = sin θ. Then sin
–1
x = θ. We have
sin
–1
(
)
2
2 1
x x
−
= sin
–1
(
)
2
2sin 1 sin
θ − θ
= sin
–1
(2sinθ cosθ) = sin
–1
(sin2θ) = 2θ
= 2 sin
–1
x
(ii) Take x = cos θ, then proceeding as above, we get, sin
–1
(
)
2
2 1
x x
−
= 2 cos
–1
x
Example 4 Show that tan
–1
–1 –1
1 2 3
tan tan
2 11 4
+ =
Solution By property 5 (i), we have
L.H.S. =
–1 –1
1 2
tan tan
2 11
+
–1 1
1 2
15
2 11
tan tan
1 2
20
1
2 11
−
+
= =
− ×
=
1
3
tan
4
−
= R.H.S.
Example 5 Express
1
cos
tan
1 sin
x
x
−
−
,
3
2 2
π π
−
< <
x
in the simplest form.
Solution We write
2 2
1 –1
2 2
cos sin
cos
2 2
tan tan
1 sin
cos sin 2sin cos
2 2 2 2
x x
x
x x x x
x
−
−
=
−
+ −
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46 MATHEMATICS
=
–1
2
cos sin cos sin
2 2 2 2
tan
cos sin
2 2
x x x x
x x
+ −
−
=
–1
cos sin
2 2
tan
cos sin
2 2
x x
x x
+
−
–1
1 tan
2
tan
1 tan
2
x
x
+
=
−
=
–1
tan tan
4 2 4 2
x x
π π
+ = +
Alternatively,
–1 –1 –1
2
sin sin
cos
2 2
tan tan tan
2
1 sin
1 cos 1 cos
2 2
x
x
x
x
x
x
π π −
−
= =
π π −
−
− − −
=
–1
2
2 2
2sin cos
4 4
tan
2
2sin
4
x x
x
π − π −
π −
=
–1
2
tan cot
4
x
π −
–1
2
tan tan
2 4
x
π π −
= −
=
–1
tan tan
4 2
x
π
+
4 2
x
π
= +
Example 6 Write
–1
2
1
cot
1
x
−
, x > 1 in the simplest form.
Solution Let x = sec θ, then
2
1
x
−
=
2
sec 1 tan
θ − = θ
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INVERSE TRIGONOMETRIC FUNCTIONS 47
Therefore,
–1
2
1
cot
1
x
−
= cot
–1
(cot
θ) = θ = sec
–1
x, which is the simplest form.
Example 7
Prove that tan
–1
x +
–1
2
2
tan
1
x
x
−
= tan
–1
3
2
3
1 3
x x
x
−
−
,
1
| |
3
x <
Solution Let x = tan θ. Then θ = tan
–1
x. We have
R.H.S. =
3 3
–1 –1
2 2
3 3tan tan
tan tan
1 3 1 3tan
x x
x
− θ− θ
=
− − θ
= tan
–1
(tan3θ) = 3θ = 3tan
–1
x = tan
–1
x + 2 tan
–1
x
= tan
–1
x + tan
–1
2
2
1
x
x
−
= L.H.S. (Why?)
Example 8 Find the value of cos (sec
–1
x + cosec
–1
x), |x| ≥ 1
Solution We have cos (sec
–1
x + cosec
–1
x) = cos
2
π
= 0
EXERCISE 2.2
Prove the following:
1. 3sin
–1
x = sin
–1
(3x – 4x
3
),
1 1
– ,
2 2
x
∈
2. 3cos
–1
x = cos
–1
(4x
3
– 3x),
1
, 1
2
x
∈
3. tan
–1
1 1
2 7 1
tan tan
11 24 2
− −
+ =
4.
1 1 1
1 1 31
2 tan tan tan
2 7 17
− − −
+ =
Write the following functions in the simplest form:
5.
2
1
1 1
tan
x
x
−
+ −
, x ≠ 0 6.
1
2
1
tan
1
x
−
−
, |x| > 1
7.
1
1 cos
tan
1 cos
x
x
−
−
+
, 0 < x < π 8.
1
cos sin
tan
cos sin
x x
x x
−
−
+
,
4
−π
< x <
3
4
π
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48 MATHEMATICS
9.
1
2 2
tan
x
a x
−
−
, |x| < a
10.
2 3
1
3 2
3
tan
3
a x x
a ax
−
−
−
, a > 0;
3 3
−
< <
a a
x
Find the values of each of the following:
11.
–1 –1
1
tan 2cos 2sin
2
12. cot (tan
–1
a + cot
–1
a)
13.
2
–1 –1
2 2
1 2 1
tan sin cos
2
1 1
x y
x y
−
+
+ +
, |x| < 1, y > 0 and xy < 1
14. If sin
–1 –1
1
sin cos 1
5
x
+ =
, then find the value of x
15. If
–1 –1
1 1
tan tan
2 2 4
x x
x x
− + π
+ =
− +
, then find the value of x
Find the values of each of the expressions in Exercises 16 to 18.
16.
–1
2
sin sin
3
π
17.
–1
3
tan tan
4
π
18.
–1 –1
3 3
tan sin cot
5 2
+
19.
1
7
cos cos is equal to
6
−
π
(A)
7
6
π
(B)
5
6
π
(C)
3
π
(D)
6
π
20.
1
1
sin sin ( )
3 2
−
π
− −
is equal to
(A)
1
2
(B)
1
3
(C)
1
4
(D) 1
21.
1 1
tan 3 cot ( 3)
− −
− −
is equal to
(A) π (B)
2
π
−
(C) 0 (D)
2 3
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INVERSE TRIGONOMETRIC FUNCTIONS 49
Miscellaneous Examples
Example 9
Find the value of
1
3
sin (sin )
5
−
π
Solution We know that
1
sin (sin )
x x
−
=
. Therefore,
1
3 3
sin (sin )
5 5
−
π π
=
But
3
,
5 2 2
π π π
∉ −
, which is the principal branch of sin
–1
x
However
3 3 2
sin ( ) sin( ) sin
5 5 5
π π π
= π − =
and
2
,
5 2 2
π π π
∈ −
Therefore
1 1
3 2 2
sin (sin ) sin (sin )
5 5 5
− −
π π π
= =
Example 10 Show that
1 1 1
3 8 84
sin sin cos
5 17 85
− − −
− =
Solution Let
1
3
sin
5
x
−
=
and
1
8
sin
17
y
−
=
Therefore
3
sin
5
x
=
and
8
sin
17
y =
Now
2
9 4
cos 1 sin 1
25 5
x x
= − = − =
(Why?)
and
2
64 15
cos 1 sin 1
289 17
y y= − = − =
We have cos (x–y) = cos x cos y + sin x sin y
=
4 15 3 8 84
5 17 5 17 85
× + × =
Therefore
1
84
cos
85
x y
−
− =
Hence
1 1 1
3 8 84
sin sin cos
5 17 85
− − −
− =
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50 MATHEMATICS
Example 11 Show that
1 1 1
12 4 63
sin cos tan
13 5 16
− − −
+ + = π
Solution Let
1 1 1
12 4 63
sin , cos , tan
13 5 16
x y z
− − −
= = =
Then
12 4 63
sin , cos , tan
13 5 16
x y z= = =
Therefore
5 3 12 3
cos , sin , tan and tan
13 5 5 4
x y x y
= = = =
We have
tan tan
tan( )
1 tan tan
x y
x y
x y
+
+ =
−
12 3
63
5 4
12 3
16
1
5 4
+
= = −
− ×
Hence
tan( ) tan
x y z
+ = −
i.e., tan (x + y) = tan (–z) or tan (x + y) = tan (π – z)
Therefore x + y = – z or x + y = π – z
Since x, y and z are positive, x + y ≠ – z (Why?)
Hence x + y + z = π or
–1 –1 –1
12 4 63
sin cos tan
13 5 16
+ + =π
Example 12 Simplify
–1
cos sin
tan
cos sin
a x b x
b x a x
−
+
, if
a
b
tan x > –1
Solution We have,
–1
cos sin
tan
cos sin
a x b x
b x a x
−
+
=
–1
cos sin
cos
tan
cos sin
cos
a x b x
b x
b x a x
b x
−
+
=
–1
tan
tan
1 tan
a
x
b
a
x
b
−
+
=
–1 –1
tan tan (tan )
a
x
b
−
=
–1
tan
a
x
b
−
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INVERSE TRIGONOMETRIC FUNCTIONS 51
Example 13 Solve tan
–1
2x + tan
–1
3x =
4
π
Solution We have tan
–1
2x + tan
–1
3x =
4
π
or
–1
2 3
tan
1 2 3
x x
x x
+
− ×
=
4
π
i.e.
–1
2
5
tan
1 6
x
x
−
=
4
π
Therefore
2
5
1 6
x
x
−
=
tan 1
4
π
=
or 6x
2
+ 5x – 1 = 0 i.e., (6x – 1) (x + 1) = 0
which gives x =
1
6
or x = – 1.
Since x = – 1 does not satisfy the equation, as the L.H.S. of the equation becomes
negative,
1
6
x
=
is the only solution of the given equation.
Miscellaneous Exercise on Chapter 2
Find the value of the following:
1.
–1
13
cos cos
6
π
2.
–1
7
tan tan
6
π
Prove that
3.
–1 –1
3 24
2sin tan
5 7
=
4.
–1 –1 –1
8 3 77
sin sin tan
17 5 36
+ =
5.
–1 –1 –1
4 12 33
cos cos cos
5 13 65
+ =
6.
–1 –1 –1
12 3 56
cos sin sin
13 5 65
+ =
7.
–1 –1 –1
63 5 3
tan sin cos
16 13 5
= +
8.
–1 1 1 1
1 1 1 1
tan tan tan tan
5 7 3 8 4
− − −
π
+ + + =
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52 MATHEMATICS
Prove that
9.
–1 –1
1 1
tan cos
2 1
x
x
x
−
=
+
, x ∈ [0, 1]
10.
–1
1 sin 1 sin
cot
2
1 sin 1 sin
x x x
x x
+ + −
=
+ − −
,
0,
4
x
π
∈
11.
–1 –1
1 1 1
tan cos
4 2
1 1
x x
x
x x
+ − − π
= −
+ + −
,
1
1
2
x
− ≤ ≤
[Hint: Put x = cos 2θ]
12.
1 1
9 9 1 9 2 2
sin sin
8 4 3 4 3
− −
π
− =
Solve the following equations:
13. 2tan
–1
(cos x) = tan
–1
(2 cosec x) 14.
–1 –1
1 1
tan tan ,( 0)
1 2
x
x x
x
−
= >
+
15. sin (tan
–1
x), |x| < 1 is equal to
(A)
2
1
x
x
−
(B)
2
1
1
x
−
(C)
2
1
1
x
+
(D)
2
1
x
x
+
16. sin
–1
(1 – x) – 2 sin
–1
x =
2
π
, then x is equal to
(A) 0,
1
2
(B) 1,
1
2
(C) 0 (D)
1
2
17.
1 1
tan tan
x x y
y x y
− −
−
−
+
is equal to
(A)
2
π
(B)
3
π
(C)
4
π
(D)
3
4
π
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INVERSE TRIGONOMETRIC FUNCTIONS 53
Summary
®
The domains and ranges (principal value branches) of inverse trigonometric
functions are given in the following table:
Functions Domain Range
(Principal Value Branches)
y = sin
–1
x [–1, 1]
,
2 2
y = cos
–1
x [–1, 1] [0, π]
y = cosec
–1
x R – (–1,1)
,
2 2
– {0}
y = sec
–1
x R – (–1, 1) [0, π] –
{ }
2
π
y = tan
–1
x R
,
2 2
π π
−
y = cot
–1
x R (0, π)
®
sin
–1
x should not be confused with (sin
x)
–1
. In fact (sin x)
–1
=
1
sin
x
and
similarly for other trigonometric functions.
®
The value of an inverse trigonometric functions which lies in its principal
value branch is called the principal value of that inverse trigonometric
functions.
For suitable values of domain, we have
®
y = sin
–1
x ⇒ x = sin y
®
x = sin y ⇒ y = sin
–1
x
®
sin (sin
–1
x) = x
®
sin
–1
(sinx) = x
®
sin
–1
1
x
= cosec
–1
x
®
cos
–1
(–x) = π – cos
–1
x
®
cos
–1
1
x
= sec
–1
x
®
cot
–1
(–x) = π – cot
–1
x
®
tan
–1
1
x
= cot
–1
x
®
sec
–1
(–x) = π – sec
–1
x
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54 MATHEMATICS
®
sin
–1
(–x) = – sin
–1
x
®
tan
–1
(–
x) = – tan
–1
x
®
tan
–1
x + cot
–1
x =
2
π
®
cosec
–1
(–x) = – cosec
–1
x
®
tan
–1
x + tan
–1
y = tan
–1
1
x y
xy
+
−
®
sin
–1
x + cos
–1
x =
2
π
®
tan
–1
x + tan
–1
y = tan
–1
1
x y
xy
+
−
®
cosec
–1
x + sec
–1
x =
2
π
®
2tan
–1
x = sin
–1
2
2
1
x
x
+
= cos
–1
2
2
1
1
x
x
−
+
®
2tan
–1
x = tan
–1
2
2
1
x
x
−
®
tan
–1
x + tan
–1
y =
π
+ tan
–1
x y
xy
+
−
1
, xy > 1; x, y > 0
Historical Note
The study of trigonometry was first started in India. The ancient Indian
Mathematicians, Aryabhata (476A.D.), Brahmagupta (598 A.D.), Bhaskara I
(600 A.D.) and Bhaskara II (1114 A.D.) got important results of trigonometry. All
this knowledge went from India to Arabia and then from there to Europe. The
Greeks had also started the study of trigonometry but their approach was so
clumsy that when the Indian approach became known, it was immediately adopted
throughout the world.
In India, the predecessor of the modern trigonometric functions, known as
the sine of an angle, and the introduction of the sine function represents one of
the main contribution of the siddhantas (Sanskrit astronomical works) to
mathematics.
Bhaskara I (about 600 A.D.) gave formulae to find the values of sine functions
for angles more than 90°. A sixteenth century Malayalam work Yuktibhasa
contains a proof for the expansion of sin (A + B). Exact expression for sines or
cosines of 18°, 36°, 54°, 72°, etc., were given by Bhaskara II.
The symbols sin
–1
x, cos
–1
x, etc., for arc sin x, arc cos x, etc., were suggested
by the astronomer Sir John F.W. Hersehel (1813) The name of Thales
(about 600 B.C.) is invariably associated with height and distance problems. He
is credited with the determination of the height of a great pyramid in Egypt by
measuring shadows of the pyramid and an auxiliary staff (or gnomon) of known
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INVERSE TRIGONOMETRIC FUNCTIONS 55
height, and comparing the ratios:
H
S
h
s
=
= tan (sun’s altitude)
Thales is also said to have calculated the distance of a ship at sea through
the proportionality of sides of similar triangles. Problems on height and distance
using the similarity property are also found in ancient Indian works.
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