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Inverse Trigonometric Functions Mathematics 12th Class Notes for CBSE 2020
- 1. 1 - Page 1 - MATHEMATICS CLASS NOTES FOR CBSE Chapter 17. Inverse Trigonometric Functions 01. Inverse of a Function We know that corresponding to every bijection (one-one onto function) → there exists a bijection → defined by if and only The function → is called the inverse of function → and is denoted by Thus, we have ⇔ We have also learnt that for all ∈ and for all ∈ 02. Inverse of Sine Function Consider the function → given by Sin It is a many-one into function as it attains same value at infinitely many points and its range is not same as its co-domain. We know that any function can be made an onto function, if we replace its co-domain by its range. Therefore, → is a many-one onto functions. ╷ x' ╷ x (0, 1) − ╷ ╷ y y' − (0,−1) y=Sinx
- 2. CLASS NOTES FOR CBSE – 17. Inverse Trigonometric Functions 2 - Page 2 - Journey to obtain inverse of Sine function- In order to make a one-one function, we will have to restrict its domain in such a way that in that domain there is no turn in the graph of the function and the function takes every value between and It is evident from the graph of Sin that if we take the domain as then becomes one-one. Thus, → given Sin is a bijection and hence invertible. The inverse of the sine function is denoted by Sin Thus, Sin is a function with domain and range such that Sin ⇔ Sin Also, Sin Sin for all ∈ ∵ and and, Sin Sin for all ∈ The graph of the function → given by Sin is shown in Figure. In order to obtain the graph of Sin → we interchange and axes as shown in Figure. ╷ ╷0 xx' y' y −(0,1) −(0,−1) y=Sinx ╷ (1,0) 0 x x' y' y −(0,− ) y=Sin−1 x (−1,0) ╷ −(0, ) 03. Principal Value Branches of Inverse Trigonometric Functions (i) Sin ⇒ Sin In Sin for one value of can take infinite values. But if Sin is a function, then should possess only one value of for every value of This means we should restrict the values which can possess. The restricted set of values which can possess is its Principal Value Branch.
- 3. CLASS NOTES FOR CBSE – 17. Inverse Trigonometric Functions 3 - Page 3 - (ii) Here ≤ Sin ≤ ⇒ ≤ ≤ ⇒ Domain: ∈ Range: ∈ Principal Value Branch of Sin ≡ ≤ ≤ 10 x y −1 (iii) Cos ⇒ Cos Here ≤ Cos ≤ ⇒ ≤ ≤ Domain: ∈ Range: ∈ Principal Value Branch of Cos ≡ ≤ ≤ 10 x y −1 (iv) Tan ⇒ Tan Here ∞ Tan ∞ ⇒ Domain: ∈ Range: ∈ Principal Value Branch of Tan ≡