Detailed Notes: Inverse Trigonometry Functions

``` Page 1

INVERSE TRIGONOMETRIC
FUNCTION
1. INTRODUCTION:
The inverse trigonometric functions, denoted by ?? ?? ?? - 1
? ?? or (arc ?? ?? ?? ? ?? ) , ?? ?? ?? - 1
? ?? etc., denote the
angles whose sine, cosine etc, is equal to ?? . The angles are usually the numerically smallest angles,
except in the case of ?? ?? ?? - 1
? ?? and if positive & negative angles have same numerical value, the
positive angle has been chosen.
It is worthwhile noting that the functions sinx, cosx etc are in general not invertible. Their inverse is
defined by choosing an appropriate domain & co-domain so that they become invertible. For this
reason the chosen value is usually the simplest and easy to remember.
2. DOMAIN & RANGE OF INVERSE
TRIGONOMETRIC FUNCTIONS:
S.No ?? ( ?? ) Domain Range
( 1 ) ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
]
( 2 ) ?? ?? ?? - 1
? ?? | ?? | = 1 [ 0 , ?? ]
( 3 ) ?? ?? ?? - 1
? ?? ?? ? ??
( -
?? 2
,
?? 2
)
(4) ?? ?? ?? - 1
? ?? | ?? | = 1
[ 0 , ?? ] - {
?? 2
} or [ 0 ,
?? 2
) ? (
?? 2
, ?? ] ]
(5) ?? ?? ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
] - { 0 }
(6) ?? ?? ?? - 1
? ?? ?? ? ?? ( 0 , ?? )

3. GRAPH OF INVERSE TRIGONOMETRIC
FUNCTIONS:
(a) ?? : [
- ?? 2
,
?? 2
] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

Page 2

INVERSE TRIGONOMETRIC
FUNCTION
1. INTRODUCTION:
The inverse trigonometric functions, denoted by ?? ?? ?? - 1
? ?? or (arc ?? ?? ?? ? ?? ) , ?? ?? ?? - 1
? ?? etc., denote the
angles whose sine, cosine etc, is equal to ?? . The angles are usually the numerically smallest angles,
except in the case of ?? ?? ?? - 1
? ?? and if positive & negative angles have same numerical value, the
positive angle has been chosen.
It is worthwhile noting that the functions sinx, cosx etc are in general not invertible. Their inverse is
defined by choosing an appropriate domain & co-domain so that they become invertible. For this
reason the chosen value is usually the simplest and easy to remember.
2. DOMAIN & RANGE OF INVERSE
TRIGONOMETRIC FUNCTIONS:
S.No ?? ( ?? ) Domain Range
( 1 ) ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
]
( 2 ) ?? ?? ?? - 1
? ?? | ?? | = 1 [ 0 , ?? ]
( 3 ) ?? ?? ?? - 1
? ?? ?? ? ??
( -
?? 2
,
?? 2
)
(4) ?? ?? ?? - 1
? ?? | ?? | = 1
[ 0 , ?? ] - {
?? 2
} or [ 0 ,
?? 2
) ? (
?? 2
, ?? ] ]
(5) ?? ?? ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
] - { 0 }
(6) ?? ?? ?? - 1
? ?? ?? ? ?? ( 0 , ?? )

3. GRAPH OF INVERSE TRIGONOMETRIC
FUNCTIONS:
(a) ?? : [
- ?? 2
,
?? 2
] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? to get ?? ?? ?? - 1
? ?? ) ?? - 1
: [ - 1 , 1 ] ? [ - ?? / 2 , ?? / 2 ]
?? - 1
( ?? ) = ?? ?? ?? - 1
? ( ?? )

( ?? = ?? ?? ?? - 1
? ?? )
(b) ?? : [ 0 , ?? ] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(c) ?? : ( - ?? / 2 , ?? / 2 ) ? ??
?? ( ?? ) = ?? ?? ?? ? ??
Page 3

INVERSE TRIGONOMETRIC
FUNCTION
1. INTRODUCTION:
The inverse trigonometric functions, denoted by ?? ?? ?? - 1
? ?? or (arc ?? ?? ?? ? ?? ) , ?? ?? ?? - 1
? ?? etc., denote the
angles whose sine, cosine etc, is equal to ?? . The angles are usually the numerically smallest angles,
except in the case of ?? ?? ?? - 1
? ?? and if positive & negative angles have same numerical value, the
positive angle has been chosen.
It is worthwhile noting that the functions sinx, cosx etc are in general not invertible. Their inverse is
defined by choosing an appropriate domain & co-domain so that they become invertible. For this
reason the chosen value is usually the simplest and easy to remember.
2. DOMAIN & RANGE OF INVERSE
TRIGONOMETRIC FUNCTIONS:
S.No ?? ( ?? ) Domain Range
( 1 ) ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
]
( 2 ) ?? ?? ?? - 1
? ?? | ?? | = 1 [ 0 , ?? ]
( 3 ) ?? ?? ?? - 1
? ?? ?? ? ??
( -
?? 2
,
?? 2
)
(4) ?? ?? ?? - 1
? ?? | ?? | = 1
[ 0 , ?? ] - {
?? 2
} or [ 0 ,
?? 2
) ? (
?? 2
, ?? ] ]
(5) ?? ?? ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
] - { 0 }
(6) ?? ?? ?? - 1
? ?? ?? ? ?? ( 0 , ?? )

3. GRAPH OF INVERSE TRIGONOMETRIC
FUNCTIONS:
(a) ?? : [
- ?? 2
,
?? 2
] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? to get ?? ?? ?? - 1
? ?? ) ?? - 1
: [ - 1 , 1 ] ? [ - ?? / 2 , ?? / 2 ]
?? - 1
( ?? ) = ?? ?? ?? - 1
? ( ?? )

( ?? = ?? ?? ?? - 1
? ?? )
(b) ?? : [ 0 , ?? ] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(c) ?? : ( - ?? / 2 , ?? / 2 ) ? ??
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(d) ?? : ( 0 , ?? ) ? ??
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(e) ? ?? : [ 0 , ?? / 2 ) ? ( ?? / 2 , ?? ] ? ( - 8 , - 1 ] ? [ 1 , 8 )
?? ( ?? ) = ?? ?? ?? ? ??
?? - 1
: ( - 8 , - 1 ] ? [ 1 , 8 ) ? [ 0 , ?? / 2 ) ? ( ?? / 2 , ?? ]
?? - 1
( ?? ) = ?? ?? ?? - 1
? ??
? ?? - 1
: [ - 1 , 1 ] ? [ 0 , ?? ] ? ? ?? - 1
( ?? ) = ?? ?? ?? - 1
? ?? ?
Page 4

INVERSE TRIGONOMETRIC
FUNCTION
1. INTRODUCTION:
The inverse trigonometric functions, denoted by ?? ?? ?? - 1
? ?? or (arc ?? ?? ?? ? ?? ) , ?? ?? ?? - 1
? ?? etc., denote the
angles whose sine, cosine etc, is equal to ?? . The angles are usually the numerically smallest angles,
except in the case of ?? ?? ?? - 1
? ?? and if positive & negative angles have same numerical value, the
positive angle has been chosen.
It is worthwhile noting that the functions sinx, cosx etc are in general not invertible. Their inverse is
defined by choosing an appropriate domain & co-domain so that they become invertible. For this
reason the chosen value is usually the simplest and easy to remember.
2. DOMAIN & RANGE OF INVERSE
TRIGONOMETRIC FUNCTIONS:
S.No ?? ( ?? ) Domain Range
( 1 ) ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
]
( 2 ) ?? ?? ?? - 1
? ?? | ?? | = 1 [ 0 , ?? ]
( 3 ) ?? ?? ?? - 1
? ?? ?? ? ??
( -
?? 2
,
?? 2
)
(4) ?? ?? ?? - 1
? ?? | ?? | = 1
[ 0 , ?? ] - {
?? 2
} or [ 0 ,
?? 2
) ? (
?? 2
, ?? ] ]
(5) ?? ?? ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
] - { 0 }
(6) ?? ?? ?? - 1
? ?? ?? ? ?? ( 0 , ?? )

3. GRAPH OF INVERSE TRIGONOMETRIC
FUNCTIONS:
(a) ?? : [
- ?? 2
,
?? 2
] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? to get ?? ?? ?? - 1
? ?? ) ?? - 1
: [ - 1 , 1 ] ? [ - ?? / 2 , ?? / 2 ]
?? - 1
( ?? ) = ?? ?? ?? - 1
? ( ?? )

( ?? = ?? ?? ?? - 1
? ?? )
(b) ?? : [ 0 , ?? ] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(c) ?? : ( - ?? / 2 , ?? / 2 ) ? ??
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(d) ?? : ( 0 , ?? ) ? ??
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(e) ? ?? : [ 0 , ?? / 2 ) ? ( ?? / 2 , ?? ] ? ( - 8 , - 1 ] ? [ 1 , 8 )
?? ( ?? ) = ?? ?? ?? ? ??
?? - 1
: ( - 8 , - 1 ] ? [ 1 , 8 ) ? [ 0 , ?? / 2 ) ? ( ?? / 2 , ?? ]
?? - 1
( ?? ) = ?? ?? ?? - 1
? ??
? ?? - 1
: [ - 1 , 1 ] ? [ 0 , ?? ] ? ? ?? - 1
( ?? ) = ?? ?? ?? - 1
? ?? ?

?? - 1
: ?? ? ( - ?? / 2 , ?? / 2 )
?? - 1
( ?? ) = ?? ?? ?? - 1
? ??

?? - 1
: ?? ? ( 0 , ?? )
?? - 1
( ?? ) = ?? ?? ?? - 1
? ??
Page 5

INVERSE TRIGONOMETRIC
FUNCTION
1. INTRODUCTION:
The inverse trigonometric functions, denoted by ?? ?? ?? - 1
? ?? or (arc ?? ?? ?? ? ?? ) , ?? ?? ?? - 1
? ?? etc., denote the
angles whose sine, cosine etc, is equal to ?? . The angles are usually the numerically smallest angles,
except in the case of ?? ?? ?? - 1
? ?? and if positive & negative angles have same numerical value, the
positive angle has been chosen.
It is worthwhile noting that the functions sinx, cosx etc are in general not invertible. Their inverse is
defined by choosing an appropriate domain & co-domain so that they become invertible. For this
reason the chosen value is usually the simplest and easy to remember.
2. DOMAIN & RANGE OF INVERSE
TRIGONOMETRIC FUNCTIONS:
S.No ?? ( ?? ) Domain Range
( 1 ) ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
]
( 2 ) ?? ?? ?? - 1
? ?? | ?? | = 1 [ 0 , ?? ]
( 3 ) ?? ?? ?? - 1
? ?? ?? ? ??
( -
?? 2
,
?? 2
)
(4) ?? ?? ?? - 1
? ?? | ?? | = 1
[ 0 , ?? ] - {
?? 2
} or [ 0 ,
?? 2
) ? (
?? 2
, ?? ] ]
(5) ?? ?? ?? ?? ?? - 1
? ?? | ?? | = 1
[ -
?? 2
,
?? 2
] - { 0 }
(6) ?? ?? ?? - 1
? ?? ?? ? ?? ( 0 , ?? )

3. GRAPH OF INVERSE TRIGONOMETRIC
FUNCTIONS:
(a) ?? : [
- ?? 2
,
?? 2
] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? to get ?? ?? ?? - 1
? ?? ) ?? - 1
: [ - 1 , 1 ] ? [ - ?? / 2 , ?? / 2 ]
?? - 1
( ?? ) = ?? ?? ?? - 1
? ( ?? )

( ?? = ?? ?? ?? - 1
? ?? )
(b) ?? : [ 0 , ?? ] ? [ - 1 , 1 ]
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(c) ?? : ( - ?? / 2 , ?? / 2 ) ? ??
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(d) ?? : ( 0 , ?? ) ? ??
?? ( ?? ) = ?? ?? ?? ? ??

(Taking image of ?? ?? ?? ? ?? about ?? = ?? )
(e) ? ?? : [ 0 , ?? / 2 ) ? ( ?? / 2 , ?? ] ? ( - 8 , - 1 ] ? [ 1 , 8 )
?? ( ?? ) = ?? ?? ?? ? ??
?? - 1
: ( - 8 , - 1 ] ? [ 1 , 8 ) ? [ 0 , ?? / 2 ) ? ( ?? / 2 , ?? ]
?? - 1
( ?? ) = ?? ?? ?? - 1
? ??
? ?? - 1
: [ - 1 , 1 ] ? [ 0 , ?? ] ? ? ?? - 1
( ?? ) = ?? ?? ?? - 1
? ?? ?

?? - 1
: ?? ? ( - ?? / 2 , ?? / 2 )
?? - 1
( ?? ) = ?? ?? ?? - 1
? ??

?? - 1
: ?? ? ( 0 , ?? )
?? - 1
( ?? ) = ?? ?? ?? - 1
? ??

(f) ?? : [ - ?? / 2 , 0 ) ? ( 0 , ?? / 2 ] ? ( - 8 , - 1 ] ? [ 1 , 8 )
? ?? ( ?? ) = ?? ?? ?? ?? ?? ? ?? ? ? ?? - 1
: ( - 8 , - 1 ] ? [ 1 , 8 ) ? [ - ?? / 2 , 0 ) ? ( 0 , ?? / 2 ] ? ? ?? - 1
( ?? ) = ?? ?? ?? ?? ?? - 1
? ?? ?

From the above discussions following IMPORTANT points can be concluded:
(i) All the inverse trigonometric functions represent an angle.
(ii) If ?? > 0, then all six inverse trigonometric functions viz
?? ?? ?? - 1
? ?? , ?? ?? ?? - 1
? ?? , ?? ?? ?? - 1
? ?? , ?? ?? ?? - 1
? ?? , ?? ?? ?? ?? ?? - 1
? ?? , ?? ?? ?? - 1
? ?? represent an acute angle.
```

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Detailed Notes: Inverse Trigonometry Functions - Mathematics (Maths) for JEE Main & Advanced

 1. What are the basic properties of inverse trigonometry functions?
Ans. The basic properties of inverse trigonometry functions include the domain, range, and principal values. The domain of inverse trigonometry functions is limited to specific intervals to ensure they have unique inverses. The range of these functions is also restricted to ensure that they have well-defined outputs. The principal values are the main values of the inverse trigonometry functions within their respective domains.
 2. How do we find the values of inverse trigonometry functions?
Ans. To find the values of inverse trigonometry functions, we use the properties of these functions along with trigonometric identities. We can also use the unit circle or right triangle to determine the values of inverse trigonometry functions. Additionally, we can use the graphs of these functions to find their values.
 3. What is the relationship between trigonometry functions and their inverses?
Ans. The relationship between trigonometry functions and their inverses is that they are reciprocals of each other. This means that the inverse trigonometry functions "undo" the actions of the trigonometry functions. For example, the sine function and its inverse, arcsine function, are reciprocals of each other.
 4. How do we solve equations involving inverse trigonometry functions?
Ans. To solve equations involving inverse trigonometry functions, we can use the properties of these functions to simplify the equations. We can also use trigonometric identities to rewrite the equations in a more manageable form. It is essential to be familiar with the properties of inverse trigonometry functions to solve such equations effectively.
 5. What are some common applications of inverse trigonometry functions in real life?
Ans. Some common applications of inverse trigonometry functions in real life include finding angles in navigation, engineering, physics, and computer graphics. Inverse trigonometry functions are used to calculate angles and distances in various practical scenarios where trigonometry is involved.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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