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CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions

CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions

CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions:-Introduction:-In Chapter 1, we have studied that the inverse

CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions

of a function f, denoted by f –1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses. In Class XI, we studied that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist. In this chapter, we shall study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behaviour through graphical representations. Besides, some elementary properties will also be discussed. The inverse trigonometric functions play an important role in calculus for they serve to define many integrals. The concepts of inverse trigonometric functions is also used in science and engineering.

CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions:-Basic Concepts

Studied trigonometric functions, which are defined as follows:

  • Sine function, i.e., sine : R→ [– 1, 1]
  • Cosine function, i.e., cos : R → [– 1, 1]
  • Tangent function, i.e., tan : R – { x : x = (2n + 1)2π, n ∈ Z} → R
  • Cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R
  • Secant function, i.e., sec : R – { x : x = (2n + 1) 2π, n ∈ Z} → R – (– 1, 1)
  • Cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)

CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions:- Properties of Inverse Trigonometric Functions

we shall prove some important properties of inverse trigonometric functions. It may be mentioned here that these results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined. Some results may not be valid for all values of the domains of inverse trigonometric functions. In fact, they will be valid only for some values of x for which inverse trigonometric functions are defined. We will not go into the details of these values of x in the domain as this discussion goes beyond the scope of this text book. Let us recall that if y = sin–1x, then x = sin y and if x = sin y, then y = sin–1x. This is equivalent to sin (sin–1 x) = x, x ∈ [– 1, 1] and sin–1 (sin x) = x, x ∈ ,2 2⎡ π π⎤ ⎢− ⎥ ⎣ ⎦ Same is true for other five inverse trigonometric functions as well. We now prove

some properties of inverse trigonometric functions.
1.

  • sin–1 1 x = cosec–1 x, x ≥ 1 or x ≤ – 1
  • cos–1 1 x = sec–1x, x ≥ 1 or x ≤ – 1
  • tan–1 1 x = cot–1 x, x > 0
    • To prove the first result, we put cosec–1 x = y, i.e., x = cosec y
      Therefore 1 x = sin y
      Hence sin–1 1 x = y
      or sin–1 1
      x = cosec–1 x
      Similarly, we can prove the other parts.

2.

(i) sin–1 (–x) = – sin–1 x, x ∈ [– 1, 1]
(ii) tan–1 (–x) = – tan–1 x, x ∈ R
(iii) cosec–1 (–x) = – cosec–1 x, | x | ≥ 1
Let sin–1 (–x) = y, i.e., –x = sin y so that x = – sin y, i.e., x = sin (–y).
Hence sin–1 x = – y = – sin–1 (–x)
Therefore sin–1 (–x) = – sin–1x
Similarly, we can prove the other parts.

3.

(i) cos–1 (–x) = π – cos–1 x, x ∈ [– 1, 1]
(ii) sec–1 (–x) = π – sec–1 x, | x | ≥ 1
(iii) cot–1 (–x) = π – cot–1 x, x ∈ R
Let cos–1 (–x) = y i.e., – x = cos y so that x = – cos y = cos (π – y)
Therefore cos–1 x = π – y = π – cos–1 (–x)
Hence cos–1 (–x) = π – cos–1 x
Similarly, we can prove the other parts.

4.

(i) sin–1 x + cos–1 x = 2π , x ∈ [– 1, 1]
(ii) tan–1 x + cot–1 x = 2π, x ∈ R
(iii) cosec–1 x + sec–1 x = 2π, | x | ≥ 1
Let sin–1 x = y. Then x = sin y = cos 2y⎛ π ⎞ ⎜ − ⎟⎝ ⎠
Therefore cos–1 x = 2yπ− = sin–12xπ−

5.

(i) tan–1x + tan–1 y = tan–1–+1x yxy, xy < 1
(ii) tan–1x – tan–1 y = tan–1+–1x yxy, xy > – 1
(iii) 2tan–1x = tan–1 21 – 2xx, | x | < 1

6.

(i) 2tan–1 x = sin–1 2 2 1+ xx , | x | ≤ 1
(ii) 2tan–1 x = cos–1221 –1+xx, x ≥ 0
(iii) 2 tan–1 x = tan–1221 –xx, – 1 < x < 1

CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions :-Summary

  • The domains and ranges (principal value branches) of inverse trigonometric
    functions are given in the following table:
FunctionsDomain Range
(Principal Value Branches)
y = sin–1 x[–1, 1][−π/2, π/2]
y = cos–1 x[–1, 1][0, π]
y = cosec–1 xR – (–1,1)[−π/2, π/2]-{0}
y = sec–1 xR – (–1, 1)[0, π] – {π/2 }
y = tan–1 xR[−π/2, π/2]
y = cot–1 xR(0, π)
  • sin–1x should not be confused with (sin x)–1. In fact (sin x)–1 =1 sin x and similarly for other trigonometric functions.
  • The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions.

For suitable values of domain, we have

  • y = sin–1 x ⇒ x = sin y
  • x = sin y ⇒ y = sin–1 x
  • sin (sin–1 x) = x
  • sin–1 (sin x) = x
  • sin–1 1 x = cosec–1 x
  • cos–1 (–x) = π – cos–1 x
  • cos–1 1 x = sec–1x
  • cot–1 (–x) = π – cot–1 x
  • tan–1 1 x = cot–1 x
  • sec–1 (–x) = π – sec–1 x
  • sin–1 (–x) = – sin–1 x
  • tan–1 (–x) = – tan–1 x
  • tan–1 x + cot–1 x =2π
  • cosec–1 (–x) = – cosec–1 x
  • sin–1 x + cos–1 x = 2π
  • cosec–1 x + sec–1 x = 2π
  • tan–1x + tan–1y = tan–1 1x yxy+−
  • 2tan–1x = tan–121 2x− x
  • tan–1x – tan–1y = tan–11x yxy−+
  • 2tan–1 x = sin–1221x+ x = cos–12211xx

CBSE Class 12 Commerce Mathematics Inverse Trigonometric Functions

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