Properties of Inverse Trigonometric Functions-Formulas, Solved Examples, Class 12 Math Chapter 2 Notes Study Material Download free pdf

What are Inverse Trigonometric Functions?

Inverse trigonometric functions are the inverse functions of the basic trigonometric functions, which are sine, cosine, tangent, cotangent, secant, and cosecant.

These functions are used to find the angle of a triangle from any of the trigonometric ratios. They are also known as arcus functions, anti-trigonometric functions, or cyclometric functions

Restriction on Trigonometric Functions

A real function in the range  ƒ : R ⇒ [-1 , 1]  defined by ƒ(x) = sin(x) is not a bijection since different images have the same image such as ƒ(0) = 0, ƒ(2π) = 0,ƒ(π) = 0, so, ƒ is not one-one. Since ƒ is not a bijection (because it is not one-one) therefore inverse does not exist.

To make a function bijective we can restrict the domain of the function to [−π/2, π/2] or [−π/2, 3π/2] or [−3π/2, 5π/2] after restriction of domain ƒ(x) = sin(x) is a bijection, therefore ƒ is invertible. i.e. to make sin(x) we can restrict it to the domain [−π/2, π/2] or [−π/2, 3π/2] or [−3π/2, 5π/2] or…….  but  [−π/2, π/2] is the Principal solution of sin θ, hence to make sin θ invertible.

Naturally, the domain [−π/2, π/2] should be considered if no other domain is mentioned. 

  • ƒ: [−π/2, π/2]
    • ⇒ [-1, 1]  is defined as ƒ(x) = sin(x) and is a bijection, hence inverse exists.
    • The inverse of sin-1 is also called arcsine and inverse functions are also called arc functions.
  • ƒ:[−π/2 , π/2] ⇒ [−1 , 1] is defined as sinθ = x ⇔ sin-1(x) = θ , θ belongs to [−π/2 , π/2] and x belongs to [−1 , 1].

Similarly, we restrict the domains of cos, tan, cot, sec, cosec so that they are invertible.

Domain and Range of Inverse Trigonometric Functions

Below are some inverse trigonometric functions with their domain and range.

FunctionDomainRange
sin-1[-1 , 1 ][−π/2 , π/2]
cos-1[-1 , 1 ][0 , π]
tan-1R[−π/2 , π/2] 
cot-1R[0 , π]
sec-1(-∞ , -1] U [1, ∞)[0 , π] − {π/2}
cosec-1(-∞ , -1] U [1 , ∞)[−π/2 , π/2] – {0}

Properties of Inverse Trigonometric Functions

There are various properties of inverse trigonometric functions that are discussed as follows:

Set 1: Properties of sin

1) sin(θ) = x  ⇔  sin-1(x) = θ , θ ∈ [ -π/2 , π/2 ], x ∈ [ -1 , 1 ]  

2) sin-1(sin(θ)) = θ , θ ∈ [ -π/2 , π/2 ]

3) sin(sin-1(x)) = x , x ∈ [ -1 , 1 ]

Examples:

  • sin(π/6) = 1/2 ⇒ sin-1(1/2) = π/6 
  • sin-1(sin(π/6)) = π/6
  • sin(sin-1(1/2)) = 1/2

Set 2: Properties of cos

4) cos(θ) = x  ⇔  cos-1(x) = θ , θ ∈ [ 0 , π ] , x ∈ [ -1 , 1 ]  

5) cos-1(cos(θ)) = θ , θ ∈ [ 0 , π ]

6) cos(cos-1(x)) = x , x ∈ [ -1 , 1 ]

Examples:

  • cos(π/6) = √3/2 ⇒ cos-1(√3/2) = π/6 
  • cos-1(cos(π/6)) = π/6
  • cos(cos-1(1/2)) = 1/2

Set 3: Properties of tan

7) tan(θ) = x  ⇔  tan-1(x) = θ , θ ∈ [ -π/2 , π/2 ] ,  x ∈ R

8) tan-1(tan(θ)) = θ , θ ∈ [ -π/2 , π/2 ]

9) tan(tan-1(x)) = x , x ∈ R

Examples:

  • tan(π/4) = 1 ⇒ tan-1(1) = π/4
  • tan-1(tan(π/4)) = π/4
  • tan(tan-1(1)) = 1

Set 4: Properties of cot

10) cot(θ) = x  ⇔  cot-1(x) = θ , θ ∈ [ 0 , π ] , x ∈ R

11) cot-1(cot(θ)) = θ , θ ∈ [ 0 , π ]

12) cot(cot-1(x)) = x , x ∈ R

Examples:

  • cot(π/4) = 1 ⇒ cot-1(1) = π/4
  • cot(cot-1(π/4)) = π/4
  • cot(cot(1)) = 1

Set 5: Properties of sec

13) sec(θ) = x ⇔ sec-1(x) = θ , θ ∈ [ 0 , π] – { π/2 } , x ∈ (-∞,-1]  ∪ [1,∞)

14) sec-1(sec(θ)) = θ , θ ∈ [ 0 , π] – { π/2 }

15) sec(sec-1(x)) = x , x ∈ ( -∞ , -1 ]  ∪ [ 1 , ∞ )

Examples:

  • sec(π/3) = 1/2 ⇒ sec-1(1/2) = π/3 
  • sec-1(sec(π/3)) = π/3
  • sec(sec-1(1/2)) = 1/2

Set 6: Properties of cosec

16) cosec(θ) = x ⇔ cosec-1(x) = θ , θ ∈ [ -π/2 , π/2 ] – { 0 } , x ∈ ( -∞ , -1 ] ∪ [ 1,∞ )

17) cosec-1(cosec(θ)) = θ , θ ∈[ -π/2 , π ] – { 0 }

18) cosec(cosec-1(x)) = x , x ∈ ( -∞,-1 ] ∪ [ 1,∞ )

Examples:

  • cosec(π/6) = 2 ⇒ cosec-1(2) = π/6 
  • cosec-1(cosec(π/6)) = π/6
  • cosec(cosec-1(2)) = 2

Set 7: Other inverse trigonometric formulas

19) sin-1(-x) = -sin-1(x) ,  x ∈ [ -1 , 1 ]  

20) cos-1(-x) = π – cos-1(x) , x ∈ [ -1 , 1 ]

21) tan-1(-x) = -tan-1(x) , x ∈ R

22) cot-1(-x) = π – cot-1(x) , x ∈ R

23) sec-1(-x) = π – sec-1(x) , x ∈ ( -∞ , -1 ] ∪ [ 1 , ∞ )

24) cosec-1(-x) = -cosec-1(x) , x ∈ ( -∞ , -1 ] ∪ [ 1 , ∞ )

Examples:

  • sin-1(-1/2) = -sin-1(1/2)
  • cos-1(-1/2) = π -cos-1(1/2)
  • tan-1(-1) =  π -tan(1)
  • cot-1(-1) = -cot-1(1)
  • sec-1(-2) = -sec-1

Set 8: Sum of two trigonometric functions

25) sin-1(x) + cos-1(x) = π/2 , x ∈ [ -1 , 1 ]

26) tan-1(x) + cot-1(x) = π/2 , x ∈ R

27) sec-1(x) + cosec-1(x) = π/2 , x ∈ ( -∞ , -1 ] ∪ [ 1 , ∞ )

Proof:

sin-1(x) + cos-1(x) = π/2 , x ∈ [ -1 , 1 ]

let sin-1(x) = y 

now, 

x = sin y = cos((π/2) − y)

⇒ cos-1(x) = (π/2) – y = (π/2) −sin-1(x)

so, sin-1(x) + cos-1(x) = π/2                                        

tan-1(x) + cot-1(x) = π/2 , x ∈ R

Let tan-1(x) = y

now, cot(π/2 − y) = x 

⇒ cot-1(x) = (π/2 − y)

tan-1(x) + cot-1(x) = y + π/2 − y

so, tan-1(x) + cot-1(x) = π/2 

Similarly, we can prove the theorem of the sum of arcsec and arccosec as well.

Set 9: Conversion of trigonometric functions 

28) sin-1(1/x) = cosec-1(x) , x≥1 or x≤−1

29) cos-1(1/x) = sec-1(x) , x ≥ 1 or x ≤ −1

30) tan-1(1/x) = −π + cot-1(x)

Proof:

sin-1(1/x) = cosec-1(x) , x ≥ 1 or x ≤ −1

let, x = cosec(y)

1/x = sin(y)

⇒ sin-1(1/x) = y

⇒ sin-1(1/x) = cosec-1(x)

Similarly, we can prove the theorem of arccos and arctan as well

Example:

sin-1(1/2) = cosec-1(2)

Set 10: Periodic functions conversion

arcsin(x) = (-1)n arcsin(x) + πn

arccos(x) = ±arc cos x + 2πn

arctan(x) = arctan(x) + πn

arccot(x) = arccot(x) + πn

where n = 0, ±1, ±2, . . .

FAQs on Properties of Inverse Trigonometric Functions

What are inverse trigonometric functions?

Inverse trigonometric functions find angles when given trigonometric values, reversing sine, cosine, tangent, etc.

What are the domains and ranges of inverse trigonometric functions?

  • Arcsine (sin⁻¹x): Domain [−1,1], Range [−π/2, π/2]
  • Arccosine (cos⁻¹x): Domain [−1,1], Range [0,π]
  • Arctangent (tan⁻¹x): Domain (−∞,∞), Range (−π/2, π/2)

What are the principal values of inverse trigonometric functions?

Principal values are the main output values within specified ranges for arcsine, arccosine, and arctangent.

What are the properties of arcsine (sin-1 x)?

Some of the properties of inverse of sin function are:

  • Domain: [−1, 1]
  • Range: [−π/2, π/2]
  • Odd Function: sin-1 (-x) = −sin-1 x

What are the properties of arccosine (cos-1 x)?

Some of the properties of inverse of cos function are:

  • Domain: [−1, 1]
  • Range: [0, π]
  • Not Odd: cos-1 (-x) = π − cos-1 x

What are the properties of arccosine (tan-1 x)?

Some of the properties of inverse of tan function are:

  • Domain: (−∞,∞)
  • Range: [−π/2, π/2]
  • Odd Function: tan-1 (-x) = − tan-1 x

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.