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Properties of Inverse Trigonometric Functions

  • Last Updated : 03 Mar, 2021

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 sinθ, hence to make sinθ 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. Below are some trigonometric functions with their domain and range.

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Function



Domain

Range

sin-1         [ -1 , 1 ]   [ −π/2 , π/2 ]
cos-1         [ -1 , 1 ]       [ 0 , π ]
tan-1              R   [ −π/2 , π/2 ] 
cot-1              R       [ 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

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, …. 




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