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# Properties of Inverse Trigonometric Functions

A real function in the range  Æ’ : R â‡’ [-1 , 1]  defined by Æ’(x) = sin(x) is not a bijection since different images have the same image such as Æ’(0) = 0, Æ’(2Ï€) = 0,Æ’(Ï€) = 0, so, Æ’ is not one-one. Since Æ’ 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 [âˆ’Ï€/2, Ï€/2] or [âˆ’Ï€/2, 3Ï€/2] or [âˆ’3Ï€/2, 5Ï€/2] after restriction of domain Æ’(x) = sin(x) is a bijection, therefore Æ’ is invertible. i.e. to make sin(x) we can restrict it to the domain [âˆ’Ï€/2, Ï€/2] or [âˆ’Ï€/2, 3Ï€/2] or [âˆ’3Ï€/2, 5Ï€/2] or…â€¦.  but  [âˆ’Ï€/2, Ï€/2] is the Principal solution of sinÎ¸, hence to make sinÎ¸ invertible. Naturally, the domain [âˆ’Ï€/2, Ï€/2] should be considered if no other domain is mentioned.

• Æ’: [âˆ’Ï€/2, Ï€/2]  â‡’ [-1, 1]  is defined as Æ’(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.
• Æ’:[âˆ’Ï€/2 , Ï€/2] â‡’ [âˆ’1 , 1] is defined as sinÎ¸ = x â‡” sin-1(x) = Î¸ , Î¸ belongs to [âˆ’Ï€/2 , Ï€/2] and x belongs to [âˆ’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.

## Properties of Inverse Trigonometric Functions

### Set 1: Properties of sin

1) sin(Î¸) = x  â‡”  sin-1(x) = Î¸ , Î¸ âˆˆ [ -Ï€/2 , Ï€/2 ], x âˆˆ [ -1 , 1 ]

2) sin-1(sin(Î¸)) = Î¸ , Î¸ âˆˆ [ -Ï€/2 , Ï€/2 ]

3) sin(sin-1(x)) = x , x âˆˆ [ -1 , 1 ]

Examples:

• sin(Ï€/6) = 1/2 â‡’ sin-1(1/2) = Ï€/6
• sin-1(sin(Ï€/6)) = Ï€/6
• sin(sin-1(1/2)) = 1/2

### Set 2: Properties of cos

4) cos(Î¸) = x  â‡”  cos-1(x) = Î¸ , Î¸ âˆˆ [ 0 , Ï€ ] , x âˆˆ [ -1 , 1 ]

5) cos-1(cos(Î¸)) = Î¸ , Î¸ âˆˆ [ 0 , Ï€ ]

6) cos(cos-1(x)) = x , x âˆˆ [ -1 , 1 ]

Examples:

• cos(Ï€/6) = âˆš3/2 â‡’ cos-1(âˆš3/2) = Ï€/6
• cos-1(cos(Ï€/6)) = Ï€/6
• cos(cos-1(1/2)) = 1/2

### Set 3: Properties of tan

7) tan(Î¸) = x  â‡”  tan-1(x) = Î¸ , Î¸ âˆˆ [ -Ï€/2 , Ï€/2 ] ,  x âˆˆ R

8) tan-1(tan(Î¸)) = Î¸ , Î¸ âˆˆ [ -Ï€/2 , Ï€/2 ]

9) tan(tan-1(x)) = x , x âˆˆ R

Examples:

• tan(Ï€/4) = 1 â‡’ tan-1(1) = Ï€/4
• tan-1(tan(Ï€/4)) = Ï€/4
• tan(tan-1(1)) = 1

### Set 4: Properties of cot

10) cot(Î¸) = x  â‡”  cot-1(x) = Î¸ , Î¸ âˆˆ [ 0 , Ï€ ] , x âˆˆ R

11) cot-1(cot(Î¸)) = Î¸ , Î¸ âˆˆ [ 0 , Ï€ ]

12) cot(cot-1(x)) = x , x âˆˆ R

Examples:

• cot(Ï€/4) = 1 â‡’ cot-1(1) = Ï€/4
• cot(cot-1(Ï€/4)) = Ï€/4
• cot(cot(1)) = 1

### Set 5: Properties of sec

13) sec(Î¸) = x â‡” sec-1(x) = Î¸ , Î¸ âˆˆ [ 0 , Ï€] – { Ï€/2 } , x âˆˆ (-âˆž,-1]  âˆª [1,âˆž)

14) sec-1(sec(Î¸)) = Î¸ , Î¸ âˆˆ [ 0 , Ï€] – { Ï€/2 }

15) sec(sec-1(x)) = x , x âˆˆ ( -âˆž , -1 ]  âˆª [ 1 , âˆž )

Examples:

• sec(Ï€/3) = 1/2 â‡’ sec-1(1/2) = Ï€/3
• sec-1(sec(Ï€/3)) = Ï€/3
• sec(sec-1(1/2)) = 1/2

### Set 6: Properties of cosec

16) cosec(Î¸) = x â‡” cosec-1(x) = Î¸ , Î¸ âˆˆ [ -Ï€/2 , Ï€/2 ] – { 0 } , x âˆˆ ( -âˆž , -1 ] âˆª [ 1,âˆž )

17) cosec-1(cosec(Î¸)) = Î¸ , Î¸ âˆˆ[ -Ï€/2 , Ï€ ] – { 0 }

18) cosec(cosec-1(x)) = x , x âˆˆ ( -âˆž,-1 ] âˆª [ 1,âˆž )

Examples:

• cosec(Ï€/6) = 2 â‡’ cosec-1(2) = Ï€/6
• cosec-1(cosec(Ï€/6)) = Ï€/6
• cosec(cosec-1(2)) = 2

### Set 7: Other inverse trigonometric formulas

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

20) cos-1(-x) = Ï€ – cos-1(x) , x âˆˆ [ -1 , 1 ]

21) tan-1(-x) = -tan-1(x) , x âˆˆ R

22) cot-1(-x) = Ï€ – cot-1(x) , x âˆˆ R

23) sec-1(-x) = Ï€ – sec-1(x) , x âˆˆ ( -âˆž , -1 ] âˆª [ 1 , âˆž )

24) cosec-1(-x) = -cosec-1(x) , x âˆˆ ( -âˆž , -1 ] âˆª [ 1 , âˆž )

Examples:

• sin-1(-1/2) = -sin-1(1/2)
• cos-1(-1/2) = Ï€ -cos-1(1/2)
• tan-1(-1) =  Ï€ -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) = Ï€/2 , x âˆˆ [ -1 , 1 ]

26) tan-1(x) + cot-1(x) = Ï€/2 , x âˆˆ R

27) sec-1(x) + cosec-1(x) = Ï€/2 , x âˆˆ ( -âˆž , -1 ] âˆª [ 1 , âˆž )

Proof:

sin-1(x) + cos-1(x) = Ï€/2 , x âˆˆ [ -1 , 1 ]

let sin-1(x) = y

now,

x = sin y = cos((Ï€/2) âˆ’ y)

â‡’ cos-1(x) = (Ï€/2) â€“ y = (Ï€/2) âˆ’sin-1(x)

so, sin-1(x) + cos-1(x) = Ï€/2

tan-1(x) + cot-1(x) = Ï€/2 , x âˆˆ R

Let tan-1(x) = y

now, cot(Ï€/2 âˆ’ y) = x

â‡’ cot-1(x) = (Ï€/2 âˆ’ y)

tan-1(x) + cot-1(x) = y + Ï€/2 âˆ’ y

so, tan-1(x) + cot-1(x) = Ï€/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) , xâ‰¥1 or xâ‰¤âˆ’1

29) cos-1(1/x) = sec-1(x) , x â‰¥ 1 or x â‰¤ âˆ’1

30) tan-1(1/x) = âˆ’Ï€ + cot-1(x)

Proof:

sin-1(1/x) = cosec-1(x) , x â‰¥ 1 or x â‰¤ âˆ’1

let, x = cosec(y)

1/x = sin(y)

â‡’ sin-1(1/x) = y

â‡’ 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) + Ï€n

arccos(x) = Â±arc cos x + 2Ï€n

arctan(x) = arctan(x) + Ï€n

arccot(x) = arccot(x) + Ï€n

where n = 0, Â±1, Â±2, â€¦.

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