25
Inverse Trigonometric
Functions
Inverse Trigonometric Function IN THIS CHAPTER ....
If a function is one-one and onto from A to B, then function g which associates Inverse Trigonometric Function
each element y Î B to one and only one element x Î A, such that y = f ( x ), then Graphs of Inverse Trigonometric
g is called the inverse function of f , denoted by x = g ( y ). Functions
Q g = f -1 Domain and Range of Inverse
\ x = f -1( y ) Trigonometric Functions
Consider the sine function with domain R and range [- 1, 1] . It is clear that Properties of Inverse
this is not a one-one and onto (bijection), so it is not invertible. So, we have to Trigonometric Functions
restrict the domain of it in such a way that it becomes one-one, then it would Sum and Difference of Inverse
become invertible. Trigonometric Function
é p pù
Suppose we consider sine as a function with domain ê - , ú and codomain
ë 2 2û
[- 1, 1 ], then it is a bijection and therefore, invertible, such types of
trigonometric functions are called inverse trigonometric functions.
Principal Value of Inverse Trigonometric Functions
Since all the trigonometric function are many one functions over their
domains, their domains and codomains needs to be restrict to make them
one-one and onto.
Lets consider the sine function. Domain of sine function is the set of all real
numbers and range is [- 1, 1]. It can be restricted to any of the intervals
[- 3p/ 2, - p/ 2], [- p/ 2, p/ 2], [p/ 2, 3p/ 2], ... etc. to make it one-one and onto.
The inverse of the sine function can be defined in each of these intervals.
Therefore, sin-1 x is a function whose domain is [- 1, 1] and the range could be
any of the intervals [- 3p/ 2, - p/ 2], [- p/ 2, p/ 2], [p/ 2, 3p/ 2], ... corresponding to
each such interval, there is a branch of the sine function. The branch with
range [- p/ 2, p/ 2] is called the principal value branch.
Hence, the domain of sin-1 x function is [- 1, 1] and range is [- p/ 2, p/ 2].
, Inverse Trigonometric Functions 609
Principal Value of Inverse Trigonometric Functions Y –1
y = sin x
S. No. Functions Interval of principal value p/2
y=x
- p /2 £ y £ p /2 ,
1. sin-1 x
where y = sin-1 x 1
0 £ y £ p,
2. cos -1 x y = sin x
where y = cos -1 x -p/2 -1
- p /2 < y < p /2 , X¢ X
3. tan-1 x 1 p/2
where y = tan-1 x
- p /2 £ y £ p /2 ,
4. cosec -1 x
where y = cosec -1 x, y ¹ 0 -1
-1 0 £ y £ p,
5. sec x
where y = sec -1 x, y ¹ p /2 -p/2
0 < y < p, Y¢
6. cot -1 x
where y = cot -1 x
Note sin -1 x is an increasing function in [–1, 1].
Note If no branch of an inverse trigonometric function is
(ii) cosec y = x Þ y = cosec-1 x ,
mentioned, then it means the principal value branch of the function.
é p ö æ pù
where y Î ê - , 0÷ È ç 0, ú
Example 1. The principal value of cos-1 (cos(2 cot -1( 2 - 1))) ë 2 ø è 2û
is equal to and x Î ( -¥ , - 1] È [1, ¥ )
2p p 3p 5p
(a) (b) (c) (d)
3 4 4 6 Y
y = cosec x
-1 -1
Sol. (c) We have, cos (cos (2 cot ( 2 - 1)))
æ æ 3p ö ö é 3p ù p/2 y=x
Þ cos-1 ç cos ç2 ´ ÷÷ êëQ cot 8 = 2 - 1úû
è è 8 øø
1
æ 3p ö 3p y = sin x
Þ cos-1 ç cos ÷= -p/2
è 4ø 4 X¢ -1
X
1 p/2
æ 5p ö -1æ 2p ö
Example 2. If tan- 1 ç tan ÷ = a , tan ç - tan ÷ = b, -1
è 4 ø è 3 ø
then -p/2
(a) 4 a - 4b = 0 (b) 4 a - 3 b = 0
(c) a > b (d) None of these Y¢
é 5p ù é æ p öù -1
Sol. (b) Clearly, tan - 1 ê tan = tan - 1 ê tan ç p + ÷ ú Note cosec x is a decreasing function in (-¥, - 1.)
ë 4 úû ë è 4 øû
It also decreases in [1, ¥ ).
æ p ö p p
= tan - 1 ç tan ÷ = Þ a = (iii) cos y = x Þ y = cos-1 x ,
è 4ø 4 4
where y Î [0, p ] and x Î [-1, 1]
é æ 2p ö ù -1 é æ p öù
and tan - 1 ê - tan ç ÷ = tan ê - tan ç p - ÷ ú
ë è 3 ø úû ë è 3 øû Y
æ pö p p p
= tan - 1 ç tan ÷ = Þ b= y = cos–1x
è 3ø 3 3
æpö æpö
Now, consider 4a - 3 b = 4 ç ÷ - 3 ç ÷ = 0 p/2 y=x
è 4ø è3ø
1
Graphs of Inverse p/2 p
Trigonometric Functions X¢
–1 y = cos x
X
(i) sin y = x Þ y = sin-1 x ,
é p pù Y¢
where y Î ê - , ú and x Î [-1, 1] -1
ë 2 2û Note cos x is a decreasing function in [-1, 1] .
Inverse Trigonometric
Functions
Inverse Trigonometric Function IN THIS CHAPTER ....
If a function is one-one and onto from A to B, then function g which associates Inverse Trigonometric Function
each element y Î B to one and only one element x Î A, such that y = f ( x ), then Graphs of Inverse Trigonometric
g is called the inverse function of f , denoted by x = g ( y ). Functions
Q g = f -1 Domain and Range of Inverse
\ x = f -1( y ) Trigonometric Functions
Consider the sine function with domain R and range [- 1, 1] . It is clear that Properties of Inverse
this is not a one-one and onto (bijection), so it is not invertible. So, we have to Trigonometric Functions
restrict the domain of it in such a way that it becomes one-one, then it would Sum and Difference of Inverse
become invertible. Trigonometric Function
é p pù
Suppose we consider sine as a function with domain ê - , ú and codomain
ë 2 2û
[- 1, 1 ], then it is a bijection and therefore, invertible, such types of
trigonometric functions are called inverse trigonometric functions.
Principal Value of Inverse Trigonometric Functions
Since all the trigonometric function are many one functions over their
domains, their domains and codomains needs to be restrict to make them
one-one and onto.
Lets consider the sine function. Domain of sine function is the set of all real
numbers and range is [- 1, 1]. It can be restricted to any of the intervals
[- 3p/ 2, - p/ 2], [- p/ 2, p/ 2], [p/ 2, 3p/ 2], ... etc. to make it one-one and onto.
The inverse of the sine function can be defined in each of these intervals.
Therefore, sin-1 x is a function whose domain is [- 1, 1] and the range could be
any of the intervals [- 3p/ 2, - p/ 2], [- p/ 2, p/ 2], [p/ 2, 3p/ 2], ... corresponding to
each such interval, there is a branch of the sine function. The branch with
range [- p/ 2, p/ 2] is called the principal value branch.
Hence, the domain of sin-1 x function is [- 1, 1] and range is [- p/ 2, p/ 2].
, Inverse Trigonometric Functions 609
Principal Value of Inverse Trigonometric Functions Y –1
y = sin x
S. No. Functions Interval of principal value p/2
y=x
- p /2 £ y £ p /2 ,
1. sin-1 x
where y = sin-1 x 1
0 £ y £ p,
2. cos -1 x y = sin x
where y = cos -1 x -p/2 -1
- p /2 < y < p /2 , X¢ X
3. tan-1 x 1 p/2
where y = tan-1 x
- p /2 £ y £ p /2 ,
4. cosec -1 x
where y = cosec -1 x, y ¹ 0 -1
-1 0 £ y £ p,
5. sec x
where y = sec -1 x, y ¹ p /2 -p/2
0 < y < p, Y¢
6. cot -1 x
where y = cot -1 x
Note sin -1 x is an increasing function in [–1, 1].
Note If no branch of an inverse trigonometric function is
(ii) cosec y = x Þ y = cosec-1 x ,
mentioned, then it means the principal value branch of the function.
é p ö æ pù
where y Î ê - , 0÷ È ç 0, ú
Example 1. The principal value of cos-1 (cos(2 cot -1( 2 - 1))) ë 2 ø è 2û
is equal to and x Î ( -¥ , - 1] È [1, ¥ )
2p p 3p 5p
(a) (b) (c) (d)
3 4 4 6 Y
y = cosec x
-1 -1
Sol. (c) We have, cos (cos (2 cot ( 2 - 1)))
æ æ 3p ö ö é 3p ù p/2 y=x
Þ cos-1 ç cos ç2 ´ ÷÷ êëQ cot 8 = 2 - 1úû
è è 8 øø
1
æ 3p ö 3p y = sin x
Þ cos-1 ç cos ÷= -p/2
è 4ø 4 X¢ -1
X
1 p/2
æ 5p ö -1æ 2p ö
Example 2. If tan- 1 ç tan ÷ = a , tan ç - tan ÷ = b, -1
è 4 ø è 3 ø
then -p/2
(a) 4 a - 4b = 0 (b) 4 a - 3 b = 0
(c) a > b (d) None of these Y¢
é 5p ù é æ p öù -1
Sol. (b) Clearly, tan - 1 ê tan = tan - 1 ê tan ç p + ÷ ú Note cosec x is a decreasing function in (-¥, - 1.)
ë 4 úû ë è 4 øû
It also decreases in [1, ¥ ).
æ p ö p p
= tan - 1 ç tan ÷ = Þ a = (iii) cos y = x Þ y = cos-1 x ,
è 4ø 4 4
where y Î [0, p ] and x Î [-1, 1]
é æ 2p ö ù -1 é æ p öù
and tan - 1 ê - tan ç ÷ = tan ê - tan ç p - ÷ ú
ë è 3 ø úû ë è 3 øû Y
æ pö p p p
= tan - 1 ç tan ÷ = Þ b= y = cos–1x
è 3ø 3 3
æpö æpö
Now, consider 4a - 3 b = 4 ç ÷ - 3 ç ÷ = 0 p/2 y=x
è 4ø è3ø
1
Graphs of Inverse p/2 p
Trigonometric Functions X¢
–1 y = cos x
X
(i) sin y = x Þ y = sin-1 x ,
é p pù Y¢
where y Î ê - , ú and x Î [-1, 1] -1
ë 2 2û Note cos x is a decreasing function in [-1, 1] .
