The document provides a comprehensive overview of inverse trigonometric functions, detailing their properties, domains, ranges, and characteristics such as being increasing or decreasing, odd or even. It discusses specific functions including sin−1, cos−1, tan−1, and their related functions, while also highlighting important identities and relationships. Additionally, it notes that none of the inverse trigonometric functions are periodic and summarizes key equations and principles related to these functions.

1. By:- Pranav Khandelwal
INVERSE
TRIGONOMETRIC
FUNCTION

2. INTRODUCTION
▶ Onto function : (aka surjective function)
- Range and Co-Domain of a function are same, for eg:
For a function to be inverse trigo function, it needs to be 1-1 and onto
function (aka bijective function)
▶ One to one function: (aka injective function)
-No two elements have same output, for eg:

3. 𝒔𝒊𝒏−𝟏 𝒙
▶ Domain : [-
1,1]
▶ Range : [-
π
,π
]
2 2
▶ Increasing
function
▶ Odd function
▶ sin−1( −𝑥) = -
sin−1 𝑥
▶ sin−1
𝑥=
𝑑 1
𝑑𝑥
1−𝑥2

4. 𝒄𝒐𝒔−𝟏 𝒙
▶ Domain : [-1,1]
▶ Range : [0, π]
▶ Decreasing function
▶ Neither odd nor even
▶ cos−1( −𝑥 )= π -
cos−1 𝑥
𝑑
𝑥
▶
𝑑
cos−1 𝑥
= -
1
1−𝑥2

5. 𝒕𝒂𝒏−𝟏 𝒙
▶ Domain : All Real
no.s π
▶ Range : (-π
,)
2
2
▶ Increasing function
▶ Odd function
▶ tan−1 −𝑥 =
−tan−1 𝑥
𝑑
𝑥
▶
𝑑
tan−1 𝑥
=
1
1+𝑥2

6. 𝒄𝒐𝒕−𝟏 𝒙
▶ Domain : All Real No.s
▶ Range : (0, π)
▶ Decreasing function
▶ Neither odd nor even
function
▶ cot−1(−𝑥)= π - cot−1 𝑥
𝑑
𝑥
▶
𝑑
cot−1(𝑥)=
−
1
1+𝑥2

7. 𝒔𝒆𝒄−𝟏 𝒙
▶ Domain : −∞, −1 𝑈 [1,
∞)
2
2
▶ Range : [0,π
) U
(𝜋
,π]
▶ Neither even nor odd
function
▶ sec−1(−𝑥) = π - sec−1 𝑥
𝑑
𝑥
▶
𝑑
sec−1 𝑥
=
1
𝑥
𝑥2−1

8. 𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙
▶ Domain : −∞. −1 U [1,
∞)
▶ Range : [
�
�
2
− , 0) U
(0,
2
𝜋
]
▶ Odd function
▶ csc−1 −𝑥=
−csc−1 𝑥
𝑑
𝑥
▶
𝑑
csc−1 𝑥
= -
1
𝑥
𝑥2−1

9. NOTE:
▶ None of the inverse trigo functions are periodic functions.
▶ 𝑓(f −1 𝑥 ) = x for every x belonging to its domain
▶ Sign of an inequality changes if we apply a decreasing function
on both sides of an inequality. Sign of inequality remains same if
it's an increasing function.
▶ 𝑦 = sin−1(sin 𝑥) = nπ + (−1)𝑛 𝑥
▶ 𝑦 = cs𝑐−1 (csc 𝑥) = nπ +
(−1)𝑛 𝑥

10. ▶ 𝑦 = cos−1 (cos 𝑥) =
2nπ ± 𝑥
▶ 𝑦 = sec−1 (sec 𝑥) =
2nπ ± 𝑥

11. ▶ 𝑦 = tan−1(tan 𝑥) =
𝑛𝜋 + 𝑥
▶ 𝑦 = cot−1(𝑐𝑜𝑡 𝑥) =
𝑛𝜋 + 𝑥

12. PROPERTIES:
▶ sin−1 𝑥 + cos−1 𝑥
=
�
�
▶ tan−1 𝑥 +
co𝑡−1 𝑥 =
2
�
�
2
▶ s𝑒𝑐−1 𝑥 +
cosec−1 𝑥 =
�
�
2
�
�
▶ sin−1 1
=
csc−1 𝑥
�
�
▶ cos−1 1
=
sec−1 𝑥
▶
tan
−1 1
𝑥
cot−1
𝑥
= ቊ
−𝜋 +
cot−1 𝑥
; 𝑥 >
0
; 𝑥 <
0
�
�
�
�
▶ tan−1 𝑥 + tan−1 1
=
ቐ2 −
𝜋
2
; 𝑥 >
0
; 𝑥 <

13. ▶ tan−1 𝑥 + tan−1 𝑦
=
tan−
1
𝑥+
𝑦
1−𝑥
𝑦
; 𝑥 > 0, 𝑦 > 0,
𝑥𝑦 < 1
𝜋 +
tan−1
𝑥+
𝑦
1−𝑥
𝑦
; 𝑥 > 0, 𝑦 > 0,
𝑥𝑦 > 1
▶ tan−1 𝑥 − tan−1 𝑦 =
tan−1
𝑥−
𝑦
1+𝑥
𝑦
; 𝑥 > 0, 𝑦
> 0

14. THANKYOU