Properties of Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, are used to find the angle that corresponds to a given trigonometric value. These functions are the inverses of the six main trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. In this article, we will explore the key properties of inverse trigonometric functions, including their domains, ranges, and unique characteristics.

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.

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, . . .

Conclusion

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