Properties of Inverse 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 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.

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, ….