# 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.

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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) = ynow,

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 ∈ RLet tan

^{-1}(x) = ynow, cot(π/2 − y) = x

⇒ cot

^{-1}(x) = (π/2 − y)tan

^{-1}(x) + cot^{-1}(x) = y + π/2 − yso, 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 ≤ −1let, 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) + πnarccos(x) = ±arc cos x + 2πn

arctan(x) = arctan(x) + πn

arccot(x) = arccot(x) + πn

where n = 0, ±1, ±2, ….