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1 | 918-921 | 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 3 Properties of Inverse Trigonometric Functions
In this section, we shall prove some important properties of inverse trigonometric
functions |
1 | 919-922 | 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 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 |
1 | 920-923 | tan–1
(
)
1
3
sec
2
−
−
−
is equal to
(A) π
(B)
−π3
(C)
π3
(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 |
1 | 921-924 | 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 |
1 | 922-925 | 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 textbook |
1 | 923-926 | 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 textbook Let us recall that if y = sin–1x, then x = sin y and if x = sin y, then y = sin–1x |
1 | 924-927 | 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 textbook Let us recall that if y = sin–1x, then x = sin y and if x = sin y, then y = sin–1x This
is equivalent to
sin (sin–1 x) = x, x ∈ [– 1, 1] and sin–1 (sin x) = x, x ∈
2,
2
−π π
For suitable values of domain similar results follow for remaining trigonometric
functions |
1 | 925-928 | We will not go into the details of these
values of x in the domain as this discussion goes beyond the scope of this textbook Let us recall that if y = sin–1x, then x = sin y and if x = sin y, then y = sin–1x This
is equivalent to
sin (sin–1 x) = x, x ∈ [– 1, 1] and sin–1 (sin x) = x, x ∈
2,
2
−π π
For suitable values of domain similar results follow for remaining trigonometric
functions Rationalised 2023-24
28
MATHEMATICS
We now consider some examples |
1 | 926-929 | Let us recall that if y = sin–1x, then x = sin y and if x = sin y, then y = sin–1x This
is equivalent to
sin (sin–1 x) = x, x ∈ [– 1, 1] and sin–1 (sin x) = x, x ∈
2,
2
−π π
For suitable values of domain similar results follow for remaining trigonometric
functions Rationalised 2023-24
28
MATHEMATICS
We now consider some examples Example 3 Show that
(i)
sin–1 (
2)
2
x1
−x
= 2 sin–1 x,
1
1
2
2
x
−
≤
≤
(ii)
sin–1 (
2)
2
x1
−x
= 2 cos–1 x,
1
1
2
≤x
≤
Solution
(i)
Let x = sin θ |
1 | 927-930 | This
is equivalent to
sin (sin–1 x) = x, x ∈ [– 1, 1] and sin–1 (sin x) = x, x ∈
2,
2
−π π
For suitable values of domain similar results follow for remaining trigonometric
functions Rationalised 2023-24
28
MATHEMATICS
We now consider some examples Example 3 Show that
(i)
sin–1 (
2)
2
x1
−x
= 2 sin–1 x,
1
1
2
2
x
−
≤
≤
(ii)
sin–1 (
2)
2
x1
−x
= 2 cos–1 x,
1
1
2
≤x
≤
Solution
(i)
Let x = sin θ Then sin–1 x = θ |
1 | 928-931 | Rationalised 2023-24
28
MATHEMATICS
We now consider some examples Example 3 Show that
(i)
sin–1 (
2)
2
x1
−x
= 2 sin–1 x,
1
1
2
2
x
−
≤
≤
(ii)
sin–1 (
2)
2
x1
−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
x1
−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
x1
−x
= 2 cos–1 x
Example 4 Express
1
cos
tan
1
sin
x
x
−
−
,
23
2
π
π
−
<
x<
in the simplest form |
1 | 929-932 | Example 3 Show that
(i)
sin–1 (
2)
2
x1
−x
= 2 sin–1 x,
1
1
2
2
x
−
≤
≤
(ii)
sin–1 (
2)
2
x1
−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
x1
−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
x1
−x
= 2 cos–1 x
Example 4 Express
1
cos
tan
1
sin
x
x
−
−
,
23
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
−
−
=
−
+
−
=
–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
+
=
−
=
tan–1
tan 4
2
4
2
x
x
π
π
+
=
+
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 29
Example 5 Write
–1
12
cot
1
x
−
, x > 1 in the simplest form |
1 | 930-933 | Then sin–1 x = θ We have
sin–1 (
2)
2
x1
−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
x1
−x
= 2 cos–1 x
Example 4 Express
1
cos
tan
1
sin
x
x
−
−
,
23
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
−
−
=
−
+
−
=
–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
+
=
−
=
tan–1
tan 4
2
4
2
x
x
π
π
+
=
+
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 29
Example 5 Write
–1
12
cot
1
x
−
, x > 1 in the simplest form Solution Let x = sec θ, then
2
1
x − =
sec2
1
θ − =tan
θ
Therefore,
–1
12
cot
1
x −
= cot–1 (cot θ) = θ = sec–1 x, which is the simplest form |
1 | 931-934 | We have
sin–1 (
2)
2
x1
−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
x1
−x
= 2 cos–1 x
Example 4 Express
1
cos
tan
1
sin
x
x
−
−
,
23
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
−
−
=
−
+
−
=
–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
+
=
−
=
tan–1
tan 4
2
4
2
x
x
π
π
+
=
+
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 29
Example 5 Write
–1
12
cot
1
x
−
, x > 1 in the simplest form Solution Let x = sec θ, then
2
1
x − =
sec2
1
θ − =tan
θ
Therefore,
–1
12
cot
1
x −
= cot–1 (cot θ) = θ = sec–1 x, which is the simplest form EXERCISE 2 |
1 | 932-935 | 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
−
−
=
−
+
−
=
–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
+
=
−
=
tan–1
tan 4
2
4
2
x
x
π
π
+
=
+
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 29
Example 5 Write
–1
12
cot
1
x
−
, x > 1 in the simplest form Solution Let x = sec θ, then
2
1
x − =
sec2
1
θ − =tan
θ
Therefore,
–1
12
cot
1
x −
= cot–1 (cot θ) = θ = sec–1 x, which is the simplest form EXERCISE 2 2
Prove the following:
1 |
1 | 933-936 | Solution Let x = sec θ, then
2
1
x − =
sec2
1
θ − =tan
θ
Therefore,
–1
12
cot
1
x −
= cot–1 (cot θ) = θ = sec–1 x, which is the simplest form EXERCISE 2 2
Prove the following:
1 3sin–1 x = sin–1 (3x – 4x3),
1
1
–
2,
2
x
∈
2 |
1 | 934-937 | EXERCISE 2 2
Prove the following:
1 3sin–1 x = sin–1 (3x – 4x3),
1
1
–
2,
2
x
∈
2 3cos–1 x = cos–1 (4x3 – 3x),
21 , 1
x
∈
Write the following functions in the simplest form:
3 |
1 | 935-938 | 2
Prove the following:
1 3sin–1 x = sin–1 (3x – 4x3),
1
1
–
2,
2
x
∈
2 3cos–1 x = cos–1 (4x3 – 3x),
21 , 1
x
∈
Write the following functions in the simplest form:
3 2
1
1
1
tan
x
x
−
+
− , x ≠ 0
4 |
1 | 936-939 | 3sin–1 x = sin–1 (3x – 4x3),
1
1
–
2,
2
x
∈
2 3cos–1 x = cos–1 (4x3 – 3x),
21 , 1
x
∈
Write the following functions in the simplest form:
3 2
1
1
1
tan
x
x
−
+
− , x ≠ 0
4 1
1
cos
tan
1
cos
x
x
−
−
+
, 0 < x < π
5 |
1 | 937-940 | 3cos–1 x = cos–1 (4x3 – 3x),
21 , 1
x
∈
Write the following functions in the simplest form:
3 2
1
1
1
tan
x
x
−
+
− , x ≠ 0
4 1
1
cos
tan
1
cos
x
x
−
−
+
, 0 < x < π
5 1 cos
sin
tan
cos
sin
x
x
x
x
−
−
+
, 4
−π < x < 3
4
π
6 |
1 | 938-941 | 2
1
1
1
tan
x
x
−
+
− , x ≠ 0
4 1
1
cos
tan
1
cos
x
x
−
−
+
, 0 < x < π
5 1 cos
sin
tan
cos
sin
x
x
x
x
−
−
+
, 4
−π < x < 3
4
π
6 1
2
2
tan
x
a
x
−
−
, |x| < a
7 |
1 | 939-942 | 1
1
cos
tan
1
cos
x
x
−
−
+
, 0 < x < π
5 1 cos
sin
tan
cos
sin
x
x
x
x
−
−
+
, 4
−π < x < 3
4
π
6 1
2
2
tan
x
a
x
−
−
, |x| < a
7 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:
8 |
1 | 940-943 | 1 cos
sin
tan
cos
sin
x
x
x
x
−
−
+
, 4
−π < x < 3
4
π
6 1
2
2
tan
x
a
x
−
−
, |x| < a
7 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:
8 –1
–1 1
tan
2cos 2sin
2
9 |
1 | 941-944 | 1
2
2
tan
x
a
x
−
−
, |x| < a
7 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:
8 –1
–1 1
tan
2cos 2sin
2
9 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
Rationalised 2023-24
30
MATHEMATICS
Find the values of each of the expressions in Exercises 16 to 18 |
1 | 942-945 | 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:
8 –1
–1 1
tan
2cos 2sin
2
9 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
Rationalised 2023-24
30
MATHEMATICS
Find the values of each of the expressions in Exercises 16 to 18 10 |
1 | 943-946 | –1
–1 1
tan
2cos 2sin
2
9 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
Rationalised 2023-24
30
MATHEMATICS
Find the values of each of the expressions in Exercises 16 to 18 10 –1
2
sin
sin 3
π
11 |
1 | 944-947 | 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
Rationalised 2023-24
30
MATHEMATICS
Find the values of each of the expressions in Exercises 16 to 18 10 –1
2
sin
sin 3
π
11 –1
3
tan
tan 4
π
12 |
1 | 945-948 | 10 –1
2
sin
sin 3
π
11 –1
3
tan
tan 4
π
12 –1
–1
3
3
tan sin
cot
5
2
+
13 |
1 | 946-949 | –1
2
sin
sin 3
π
11 –1
3
tan
tan 4
π
12 –1
–1
3
3
tan sin
cot
5
2
+
13 1
7
cos
cos
is equal to
6
−
π
(A) 7
π6
(B) 5
π6
(C)
π3
(D)
6
π
14 |
1 | 947-950 | –1
3
tan
tan 4
π
12 –1
–1
3
3
tan sin
cot
5
2
+
13 1
7
cos
cos
is equal to
6
−
π
(A) 7
π6
(B) 5
π6
(C)
π3
(D)
6
π
14 1
1
sin
sin
(
)
3
2
−
π
−
−
is equal to
(A) 1
2
(B) 1
3
(C) 1
4
(D) 1
15 |
1 | 948-951 | –1
–1
3
3
tan sin
cot
5
2
+
13 1
7
cos
cos
is equal to
6
−
π
(A) 7
π6
(B) 5
π6
(C)
π3
(D)
6
π
14 1
1
sin
sin
(
)
3
2
−
π
−
−
is equal to
(A) 1
2
(B) 1
3
(C) 1
4
(D) 1
15 1
1
tan
3
cot
(
3)
−
−
−
−
is equal to
(A) π
(B)
−π2
(C) 0
(D) 2 3
Miscellaneous Examples
Example 6 Find the value of
1
3
sin
(sin
)
5
−
π
Solution We know that
sin1
(sin )
x
x
−
= |
1 | 949-952 | 1
7
cos
cos
is equal to
6
−
π
(A) 7
π6
(B) 5
π6
(C)
π3
(D)
6
π
14 1
1
sin
sin
(
)
3
2
−
π
−
−
is equal to
(A) 1
2
(B) 1
3
(C) 1
4
(D) 1
15 1
1
tan
3
cot
(
3)
−
−
−
−
is equal to
(A) π
(B)
−π2
(C) 0
(D) 2 3
Miscellaneous Examples
Example 6 Find the value of
1
3
sin
(sin
)
5
−
π
Solution We know that
sin1
(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
=
=
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 31
Miscellaneous Exercise on Chapter 2
Find the value of the following:
1 |
1 | 950-953 | 1
1
sin
sin
(
)
3
2
−
π
−
−
is equal to
(A) 1
2
(B) 1
3
(C) 1
4
(D) 1
15 1
1
tan
3
cot
(
3)
−
−
−
−
is equal to
(A) π
(B)
−π2
(C) 0
(D) 2 3
Miscellaneous Examples
Example 6 Find the value of
1
3
sin
(sin
)
5
−
π
Solution We know that
sin1
(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
=
=
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 31
Miscellaneous Exercise on Chapter 2
Find the value of the following:
1 –1
13
cos
cos 6
π
2 |
1 | 951-954 | 1
1
tan
3
cot
(
3)
−
−
−
−
is equal to
(A) π
(B)
−π2
(C) 0
(D) 2 3
Miscellaneous Examples
Example 6 Find the value of
1
3
sin
(sin
)
5
−
π
Solution We know that
sin1
(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
=
=
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 31
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 | 952-955 | 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
=
=
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 31
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 | 953-956 | –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 | 954-957 | –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 | 955-958 | –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 | 956-959 | –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
=
+
Prove that
8 |
1 | 957-960 | –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
=
+
Prove that
8 –1
–1
1
1
tan
2cos
1
x
x
x
−
=
+
, x ∈ [0, 1]
9 |
1 | 958-961 | –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
=
+
Prove that
8 –1
–1
1
1
tan
2cos
1
x
x
x
−
=
+
, x ∈ [0, 1]
9 –1
1
sin
1
sin
cot
2
1
sin
1
sin
x
x
x
x
x
+
+
−
=
+
−
−
,
0, 4
x
π
∈
10 |
1 | 959-962 | –1
–1
–1
63
5
3
tan
sin
cos
16
13
5
=
+
Prove that
8 –1
–1
1
1
tan
2cos
1
x
x
x
−
=
+
, x ∈ [0, 1]
9 –1
1
sin
1
sin
cot
2
1
sin
1
sin
x
x
x
x
x
+
+
−
=
+
−
−
,
0, 4
x
π
∈
10 –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]
Solve the following equations:
11 |
1 | 960-963 | –1
–1
1
1
tan
2cos
1
x
x
x
−
=
+
, x ∈ [0, 1]
9 –1
1
sin
1
sin
cot
2
1
sin
1
sin
x
x
x
x
x
+
+
−
=
+
−
−
,
0, 4
x
π
∈
10 –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]
Solve the following equations:
11 2tan–1 (cos x) = tan–1 (2 cosec x)
12 |
1 | 961-964 | –1
1
sin
1
sin
cot
2
1
sin
1
sin
x
x
x
x
x
+
+
−
=
+
−
−
,
0, 4
x
π
∈
10 –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]
Solve the following equations:
11 2tan–1 (cos x) = tan–1 (2 cosec x)
12 –1
–1
1
1
tan
tan
,(
0)
1
2
x
x x
x
−
=
>
+
13 |
1 | 962-965 | –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]
Solve the following equations:
11 2tan–1 (cos x) = tan–1 (2 cosec x)
12 –1
–1
1
1
tan
tan
,(
0)
1
2
x
x x
x
−
=
>
+
13 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
+
14 |
1 | 963-966 | 2tan–1 (cos x) = tan–1 (2 cosec x)
12 –1
–1
1
1
tan
tan
,(
0)
1
2
x
x x
x
−
=
>
+
13 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
+
14 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
Rationalised 2023-24
32
MATHEMATICS
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
−π π
y = cos–1 x
[–1, 1]
[0, π]
y = cosec–1 x
R – (–1,1)
2,
2
−π π
– {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–1x should not be confused with (sin x)–1 |
1 | 964-967 | –1
–1
1
1
tan
tan
,(
0)
1
2
x
x x
x
−
=
>
+
13 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
+
14 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
Rationalised 2023-24
32
MATHEMATICS
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
−π π
y = cos–1 x
[–1, 1]
[0, π]
y = cosec–1 x
R – (–1,1)
2,
2
−π π
– {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–1x should not be confused with (sin x)–1 In fact (sin x)–1 =
1
sin x and
similarly for other trigonometric functions |
1 | 965-968 | 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
+
14 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
Rationalised 2023-24
32
MATHEMATICS
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
−π π
y = cos–1 x
[–1, 1]
[0, π]
y = cosec–1 x
R – (–1,1)
2,
2
−π π
– {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–1x 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 |
1 | 966-969 | 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
Rationalised 2023-24
32
MATHEMATICS
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
−π π
y = cos–1 x
[–1, 1]
[0, π]
y = cosec–1 x
R – (–1,1)
2,
2
−π π
– {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–1x 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 (sin x) = x
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 33
Historical Note
The study of trigonometry was first started in India |
1 | 967-970 | 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 (sin x) = x
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 33
Historical Note
The study of trigonometry was first started in India The ancient Indian
Mathematicians, Aryabhata (476A |
1 | 968-971 | ® 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 (sin x) = x
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 33
Historical Note
The study of trigonometry was first started in India The ancient Indian
Mathematicians, Aryabhata (476A D |
1 | 969-972 | 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 (sin x) = x
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 33
Historical Note
The study of trigonometry was first started in India The ancient Indian
Mathematicians, Aryabhata (476A D ), Brahmagupta (598 A |
1 | 970-973 | The ancient Indian
Mathematicians, Aryabhata (476A D ), Brahmagupta (598 A D |
1 | 971-974 | D ), Brahmagupta (598 A D ), Bhaskara I
(600 A |
1 | 972-975 | ), Brahmagupta (598 A D ), Bhaskara I
(600 A D |
1 | 973-976 | D ), Bhaskara I
(600 A D ) and Bhaskara II (1114 A |
1 | 974-977 | ), Bhaskara I
(600 A D ) and Bhaskara II (1114 A D |
1 | 975-978 | D ) and Bhaskara II (1114 A D ) got important results of trigonometry |
1 | 976-979 | ) 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 |
1 | 977-980 | 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 |
1 | 978-981 | ) 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 |
1 | 979-982 | 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 |
1 | 980-983 | 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 |
1 | 981-984 | 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° |
1 | 982-985 | 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) |
1 | 983-986 | 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 |
1 | 984-987 | ) 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 |
1 | 985-988 | 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 |
1 | 986-989 | 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 |
1 | 987-990 | , 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 |
1 | 988-991 | 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 |
1 | 989-992 | , 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 |
1 | 990-993 | , were suggested
by the astronomer Sir John F W Hersehel (1813) The name of Thales
(about 600 B C |
1 | 991-994 | W Hersehel (1813) The name of Thales
(about 600 B C ) is invariably associated with height and distance problems |
1 | 992-995 | 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
height, and comparing the ratios:
H
S
=sh
= 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 |
1 | 993-996 | 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
height, and comparing the ratios:
H
S
=sh
= 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 |
1 | 994-997 | ) 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
height, and comparing the ratios:
H
S
=sh
= 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 —v
v
v
v
v—
Rationalised 2023-24
34
MATHEMATICS
vThe essence of Mathematics lies in its freedom |
1 | 995-998 | 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
height, and comparing the ratios:
H
S
=sh
= 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 —v
v
v
v
v—
Rationalised 2023-24
34
MATHEMATICS
vThe essence of Mathematics lies in its freedom — CANTOR v
3 |
1 | 996-999 | Problems on height and distance
using the similarity property are also found in ancient Indian works —v
v
v
v
v—
Rationalised 2023-24
34
MATHEMATICS
vThe essence of Mathematics lies in its freedom — CANTOR v
3 1 Introduction
The knowledge of matrices is necessary in various branches of mathematics |
1 | 997-1000 | —v
v
v
v
v—
Rationalised 2023-24
34
MATHEMATICS
vThe essence of Mathematics lies in its freedom — CANTOR v
3 1 Introduction
The knowledge of matrices is necessary in various branches of mathematics Matrices
are one of the most powerful tools in mathematics |
1 | 998-1001 | — CANTOR v
3 1 Introduction
The knowledge of matrices is necessary in various branches of mathematics Matrices
are one of the most powerful tools in mathematics This mathematical tool simplifies
our work to a great extent when compared with other straight forward methods |
1 | 999-1002 | 1 Introduction
The knowledge of matrices is necessary in various branches of mathematics Matrices
are one of the most powerful tools in mathematics This mathematical tool simplifies
our work to a great extent when compared with other straight forward methods The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations |
1 | 1000-1003 | Matrices
are one of the most powerful tools in mathematics This mathematical tool simplifies
our work to a great extent when compared with other straight forward methods The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations Matrices are not only used as a
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use |
1 | 1001-1004 | This mathematical tool simplifies
our work to a great extent when compared with other straight forward methods The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations Matrices are not only used as a
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use Matrix notation and operations are used in electronic spreadsheet
programs for personal computer, which in turn is used in different areas of business
and science like budgeting, sales projection, cost estimation, analysing the results of an
experiment etc |
1 | 1002-1005 | The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations Matrices are not only used as a
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use Matrix notation and operations are used in electronic spreadsheet
programs for personal computer, which in turn is used in different areas of business
and science like budgeting, sales projection, cost estimation, analysing the results of an
experiment etc Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices |
1 | 1003-1006 | Matrices are not only used as a
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use Matrix notation and operations are used in electronic spreadsheet
programs for personal computer, which in turn is used in different areas of business
and science like budgeting, sales projection, cost estimation, analysing the results of an
experiment etc Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices Matrices
are also used in cryptography |
1 | 1004-1007 | Matrix notation and operations are used in electronic spreadsheet
programs for personal computer, which in turn is used in different areas of business
and science like budgeting, sales projection, cost estimation, analysing the results of an
experiment etc Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices Matrices
are also used in cryptography This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
management |
1 | 1005-1008 | Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices Matrices
are also used in cryptography This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
management In this chapter, we shall find it interesting to become acquainted with the
fundamentals of matrix and matrix algebra |
1 | 1006-1009 | Matrices
are also used in cryptography This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
management In this chapter, we shall find it interesting to become acquainted with the
fundamentals of matrix and matrix algebra 3 |
1 | 1007-1010 | This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
management In this chapter, we shall find it interesting to become acquainted with the
fundamentals of matrix and matrix algebra 3 2 Matrix
Suppose we wish to express the information that Radha has 15 notebooks |
1 | 1008-1011 | In this chapter, we shall find it interesting to become acquainted with the
fundamentals of matrix and matrix algebra 3 2 Matrix
Suppose we wish to express the information that Radha has 15 notebooks We may
express it as [15] with the understanding that the number inside [ ] is the number of
notebooks that Radha has |
1 | 1009-1012 | 3 2 Matrix
Suppose we wish to express the information that Radha has 15 notebooks We may
express it as [15] with the understanding that the number inside [ ] is the number of
notebooks that Radha has Now, if we have to express that Radha has 15 notebooks
and 6 pens |
1 | 1010-1013 | 2 Matrix
Suppose we wish to express the information that Radha has 15 notebooks We may
express it as [15] with the understanding that the number inside [ ] is the number of
notebooks that Radha has Now, if we have to express that Radha has 15 notebooks
and 6 pens We may express it as [15 6] with the understanding that first number
inside [ ] is the number of notebooks while the other one is the number of pens possessed
by Radha |
1 | 1011-1014 | We may
express it as [15] with the understanding that the number inside [ ] is the number of
notebooks that Radha has Now, if we have to express that Radha has 15 notebooks
and 6 pens We may express it as [15 6] with the understanding that first number
inside [ ] is the number of notebooks while the other one is the number of pens possessed
by Radha Let us now suppose that we wish to express the information of possession
Chapter 3
MATRICES
Rationalised 2023-24
MATRICES 35
of notebooks and pens by Radha and her two friends Fauzia and Simran which
is as follows:
Radha
has
15
notebooks
and
6 pens,
Fauzia
has
10
notebooks
and
2 pens,
Simran
has
13
notebooks
and
5 pens |
1 | 1012-1015 | Now, if we have to express that Radha has 15 notebooks
and 6 pens We may express it as [15 6] with the understanding that first number
inside [ ] is the number of notebooks while the other one is the number of pens possessed
by Radha Let us now suppose that we wish to express the information of possession
Chapter 3
MATRICES
Rationalised 2023-24
MATRICES 35
of notebooks and pens by Radha and her two friends Fauzia and Simran which
is as follows:
Radha
has
15
notebooks
and
6 pens,
Fauzia
has
10
notebooks
and
2 pens,
Simran
has
13
notebooks
and
5 pens Now this could be arranged in the tabular form as follows:
Notebooks
Pens
Radha
15
6
Fauzia
10
2
Simran
13
5
and this can be expressed as
or
Radha
Fauzia
Simran
Notebooks
15
10
13
Pens
6
2
5
which can be expressed as:
In the first arrangement the entries in the first column represent the number of
note books possessed by Radha, Fauzia and Simran, respectively and the entries in the
second column represent the number of pens possessed by Radha, Fauzia and Simran,
Rationalised 2023-24
36
MATHEMATICS
respectively |
1 | 1013-1016 | We may express it as [15 6] with the understanding that first number
inside [ ] is the number of notebooks while the other one is the number of pens possessed
by Radha Let us now suppose that we wish to express the information of possession
Chapter 3
MATRICES
Rationalised 2023-24
MATRICES 35
of notebooks and pens by Radha and her two friends Fauzia and Simran which
is as follows:
Radha
has
15
notebooks
and
6 pens,
Fauzia
has
10
notebooks
and
2 pens,
Simran
has
13
notebooks
and
5 pens Now this could be arranged in the tabular form as follows:
Notebooks
Pens
Radha
15
6
Fauzia
10
2
Simran
13
5
and this can be expressed as
or
Radha
Fauzia
Simran
Notebooks
15
10
13
Pens
6
2
5
which can be expressed as:
In the first arrangement the entries in the first column represent the number of
note books possessed by Radha, Fauzia and Simran, respectively and the entries in the
second column represent the number of pens possessed by Radha, Fauzia and Simran,
Rationalised 2023-24
36
MATHEMATICS
respectively Similarly, in the second arrangement, the entries in the first row represent
the number of notebooks possessed by Radha, Fauzia and Simran, respectively |
1 | 1014-1017 | Let us now suppose that we wish to express the information of possession
Chapter 3
MATRICES
Rationalised 2023-24
MATRICES 35
of notebooks and pens by Radha and her two friends Fauzia and Simran which
is as follows:
Radha
has
15
notebooks
and
6 pens,
Fauzia
has
10
notebooks
and
2 pens,
Simran
has
13
notebooks
and
5 pens Now this could be arranged in the tabular form as follows:
Notebooks
Pens
Radha
15
6
Fauzia
10
2
Simran
13
5
and this can be expressed as
or
Radha
Fauzia
Simran
Notebooks
15
10
13
Pens
6
2
5
which can be expressed as:
In the first arrangement the entries in the first column represent the number of
note books possessed by Radha, Fauzia and Simran, respectively and the entries in the
second column represent the number of pens possessed by Radha, Fauzia and Simran,
Rationalised 2023-24
36
MATHEMATICS
respectively Similarly, in the second arrangement, the entries in the first row represent
the number of notebooks possessed by Radha, Fauzia and Simran, respectively The
entries in the second row represent the number of pens possessed by Radha, Fauzia
and Simran, respectively |
1 | 1015-1018 | Now this could be arranged in the tabular form as follows:
Notebooks
Pens
Radha
15
6
Fauzia
10
2
Simran
13
5
and this can be expressed as
or
Radha
Fauzia
Simran
Notebooks
15
10
13
Pens
6
2
5
which can be expressed as:
In the first arrangement the entries in the first column represent the number of
note books possessed by Radha, Fauzia and Simran, respectively and the entries in the
second column represent the number of pens possessed by Radha, Fauzia and Simran,
Rationalised 2023-24
36
MATHEMATICS
respectively Similarly, in the second arrangement, the entries in the first row represent
the number of notebooks possessed by Radha, Fauzia and Simran, respectively The
entries in the second row represent the number of pens possessed by Radha, Fauzia
and Simran, respectively An arrangement or display of the above kind is called a
matrix |
1 | 1016-1019 | Similarly, in the second arrangement, the entries in the first row represent
the number of notebooks possessed by Radha, Fauzia and Simran, respectively The
entries in the second row represent the number of pens possessed by Radha, Fauzia
and Simran, respectively An arrangement or display of the above kind is called a
matrix Formally, we define matrix as:
Definition 1 A matrix is an ordered rectangular array of numbers or functions |
1 | 1017-1020 | The
entries in the second row represent the number of pens possessed by Radha, Fauzia
and Simran, respectively An arrangement or display of the above kind is called a
matrix Formally, we define matrix as:
Definition 1 A matrix is an ordered rectangular array of numbers or functions The
numbers or functions are called the elements or the entries of the matrix |
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