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INVERSE TRIGONOMETRIC FUNCTIONS

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What are inverse trigonometric functions?  

in_trigno_1The inverse trigonometric functions are the inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions.



Domain and Range of Inverse Trigonometric Functions

Function Name                Domain                  Range
1. Arc Sine                     -1 ≤ x ≤ 1           - Π2 ≤ y ≤ Π2
2. Arc Cosine                  -1 ≤ x ≤ 1             0 ≤ y ≤ Π
3. Arc Tangent             All real numbers       - Π2 ≤ y ≤ Π2
4. Arc Cosecant             x ≤ −1 or 1 ≤ x      - Π2 ≤ y < 0 or 0 < y ≤ Π2
5. Arc Secant                x ≤ −1 or 1 ≤ x        0 ≤ y < Π2 or Π2 < y ≤ Π
6. Arc Cotangent          All real numbers        0 < y < Π


Graphs of Inverse Trigonometric Functions 

As discussed above, the domain and range of all inverse trigonometric functions, we shall now represent each of them on graph.
                                       in_trigno_2
                Sin-1x                           Cos-1x                                      Tan-1x



                       Cosec-1x                                                Sec-1x

                                   in_trigno_3


                                                      Cot-1x

                              in_trigno_5


Properties of Inverse Trigonometric Functions

1. (i) Sin-11x= Cosec-1x, x ≥ 1 or x ≤ -1

Proof: Let Cosec-1x = y, that is x = Cosec y

1x = Sin y

Sin-11x = y

Sin-11x = Cosec-1x


(ii) Cos-11x = Sec-1x, x ≥ 1 or x ≤ -1

Proof: Let Sec-1x = y, that is x = Sec y

1x = Cos y

Cos-11x = y

Cos-11x = Sec-1x


(iii) Tan-11x = Cot-1x, x > 0

Proof: Let Cot-1x = y that is x = Cot y

1x = Tan y

Tan-11x = y

Tan-11x = Cot-1x

2. (i) Sin-1(-x) = - Sin-1x, x ε [-1, 1]

Proof: Let Sin-1(-x) = y

-x = Sin y

x = - Sin y

x = Sin (-y)

Sin-1x = -y = - Sin-1(-x)

Sin-1(-x) = - Sin-1x

(ii) Tan-1 (–x) = – Tan-1 x, x ∈ R

Proof: Let Tan-1(-x) = y

-x = Tan y

x = - Tan y

x = Tan (-y)

Tan-1x = -y = - Tan-1(-x)

Tan-1(-x) = - Tan-1x


(iii) Cosec-1 (–x) = – Cosec-1 x, | x | ≥ 1

Proof: Let Cosec-1(-x) = y

-x = Cosec y

x = - Cosec y

x = Cosec (-y)

Cosec-1x = -y = - Cosec-1(-x)

Cosec-1(-x) = - Cosec-1x


3. (i) Cos-1 (–x) = π – Cos-1 x, x ∈ [– 1, 1]

Proof: Let Cos-1 (–x) = y i.e., – x = Cos y so that x = – Cos y = Cos (π – y)

Cos-1 x = π – y = π – Cos-1 (–x)

Hence Cos-1 (–x) = π – Cos-1 x


(ii) Sec-1 (–x) = π – Sec-1 x, | x | ≥ 1

Proof: Let Sec-1 (–x) = y i.e., – x = Sec y so that x = – Sec y = Sec (π – y)

Sec-1 x = π – y = π – Sec-1 (–x)

Hence Sec-1 (–x) = π – Sec-1 x


(iii) Cot-1 (–x) = π – Cot-1 x, x ∈ R

Proof: Let Cot-1 (–x) = y i.e., – x = Cot y so that x = – Cot y = Cot (π – y)

Cot-1 x = π – y = π – Cot-1 (–x)

Hence Cot-1 (–x) = π – Cot-1 x


4. (i) Sin-1 x + Cos-1 x = Π2 , x ∈ [– 1, 1]

Let Sin-1 x = y. Then x = Sin y = Cos (Π2 – y)

Cos-1 x = Π2 – y

Cos-1 x = Π2 – Sin-1 x
Hence, Sin-1 x + Cos-1 x = Π2


(ii) Tan-1 x + Cot-1 x =  Π2 , x ∈ R

Let Tan-1 x = y. Then x = Tan y = Cot (Π2 – y)

Cot-1 x = Π2 – y

Cot-1 x = Π2 – Tan-1 x

Tan-1 x + Cot-1 x =  Π2


(iii) Cosec-1 x + Sec-1 x =   Π2 ,| x | ≥ 1

Let Cosec-1 x = y. Then x = Cosec y = Sec (Π2 – y)

Sec-1 x = Π2 – y

Sec-1 x = Π2 – Cosec-1 x

Sec-1 x + Cosec-1 x =  Π2


5.  Tan x-1 + Tan-1 y = Tan-1 x + y , xy < 1
                                              1 - xy

Proof: Let Tan-1x = θ and Tan-1y = Ф

x = Tan θ and y = Tan Ф


Now Tan (θ + Ф) = Tan θ + Tan Ф

                            1 – Tan θ Tan Ф


Tan (θ + Ф) = x + y
                     1 – xy


θ + Ф = Tan-1 x + y
                     1 – xy

Tan x-1 + Tan-1 y = Tan-1 x + y – (I)
                                     1 - xy


In the above equation (I) replace y by (-y), we get another result:

Tan x-1 - Tan-1 y = Tan-1 x – y, xy > -1
                                       1 + xy


If we replace y by x in equation (I) we get:

2 Tan-1x = Tan-1 2x    , │x│< 1
                         1 – x2


6. (i) 2 Tan-1x = Sin-1 2x     , │x│≤ 1
                                 1 + x2


(ii) 2 Tan-1x = Cos-1 1 – x2   , x ≥ 0
                                1 + x2


(iii) 2 Tan-1x = Tan-1 2x      , -1 < x < 1
                              1 - x2



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