What are inverse trigonometric 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.
Sin-1x Cos-1x Tan-1x
Cosec-1x Sec-1x
Cot-1x
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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