CBSE Class 12 – Page 44

By going through these CBSE Class 12 Maths Notes Chapter 2 Inverse Trigonometric Functions, students can recall all the concepts quickly.

Inverse Trigonometric Functions Notes Class 12 Maths Chapter 2

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As we have learned in class XI, the domain and range of trigonometric functions are given below:
Functions Domain Range
Inverse Trigonometric Functions Class 12 Notes Maths 1
→ Inverse Function: We know that if f: X →Y such that y = f(x) is one-one and onto, then we define another function g : Y → X such that x = g(y), where x ∈ X and y ∈ Y, which is also one-one and onto.

In such a case, Domain of g = Range of f and Range of g = Domain of f.
g is called the inverse of f or g = f^-1.
∴ Inverse of g = g^-1 = (f^-1)^-1 =f.

The graph of the sin^-1 function is shown here.

Principal Value Branch of Function sin^-1:
It may be noted that for the domain [-1,1}, the range could be any one of the intervals. …..,
[\(\frac{-3 \pi}{2}\), \(\frac{-\pi}{2}\)], [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)] or [\(\frac{\pi}{2}\), \(\frac{3\pi}{2}\)] …….
Inverse Trigonometric Functions Class 12 Notes Maths 2
Corresponding to each interval, we get a branch of the function sin^-1.
The branch with range [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)] is called the principal value branch.
Thus, sin^-1: [-1,1] → [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)]

Principal Value Branch of Function cos^-1:
The graph of the function cos^-1 is as shown here.
Inverse Trigonometric Functions Class 12 Notes Maths 3
The domain of the function cos^-1 is [-1,1]. Its range is one of the intervals. ………., (-π, 0), (0, π), (π, 2π), ………… It is one-one and onto with the range [-1, 1]. The branch with range (0, π) is called the principal value branch of the function cos-1. Thus, cos-1: [-1,1] → [0, π].

Principal Value Branch of Function tan^-1:
The function tan^-1 is defined whose domain is set of real numbers and range is one of the intervais. [ \(\frac{-3 \pi}{2}\), \(\frac{-\pi}{2}\)], [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)] or [\(\frac{\pi}{2}\), \(\frac{3 \pi}{2}\)]…………

The graph of the function is as shown in the adjoining figure.
Inverse Trigonometric Functions Class 12 Notes Maths 4
The branch with range [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)] is called the principal value branch of function tan1. Thus, tan^-1: R → [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)].

Principal Value Branch of Function cosec^-1:
The graph of function cosec^-1 is as shown here. The function cosec^-1 defined whose domain is R — (-1, 1) and the range is any one of the intervals …….., [ \(\frac{-3 \pi}{2}\), \(\frac{-\pi}{2}\)] – {π}, [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)] – {0}, [\(\frac{\pi}{2}\), \(\frac{3 \pi}{2}\)] – {π}……..
Inverse Trigonometric Functions Class 12 Notes Maths 5
The function corresponding to the range [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)] – {0} is called the principal value branch of cosec^-1.
Thus, cosec^-1: R – (-1, 1) → [\(\frac{-\pi}{2}\), \(\frac{\pi}{2}\)] – {0}.

Principal Value Branch of Function sec^-1:
The graph of function sec^-1 is shown in adjoining figure. The sec^-1 is defined as a function whose domain is R – (-1, 1) and the range is one of the intervals……. [- π, 0] – {\(\frac{-\pi}{2}\)}, [0, π] – {\(\frac{\pi}{2}\)}, [π, 2π] – {\(\frac{3 \pi}{2}\)}, ……………
Inverse Trigonometric Functions Class 12 Notes Maths 6
The function corresponding to [0, π] – {\(\frac{\pi}{2}\)} is known as the range [0, π] – {\(\frac{\pi}{2}\)} is known as the principal value branch of sec-1.

Thus, sec^-1: R – (-1,1) → [0, π] – {\(\frac{\pi}{2}\)}

Principal Value Branch of Function cot^-1:
The graph of function cor’ is as shown here.
Inverse Trigonometric Functions Class 12 Notes Maths 7
The cot^-1 function is defined as a function whose domain is R and the range is any of the intervals, …………
(-π, 0), (0, π), (π, 2π), …………

The function corresponding to (0, π) is called the principal value branch of the function cot^-1.
Thus, cot^-1: R → (0, π).

The principal value branch of trigonometric inverse functions are as follows:
Inverse Trigonometric Functions Class 12 Notes Maths 8
→ Some Important Results:
1. (a) sin^-1\(\frac{1}{x}\) = cosec^-1x, x ≥ 1, x ≤ -1
(b) cos^-1 \(\frac{1}{x}\) = sec^-1x, x ≥ l, x ≤ -1
(c) tan^-1 \(\frac{1}{x}\) = cot^-1x, x > 0
(d) cosec^-1 \(\frac{1}{x}\) = sin^-1x, x ∈ [-1,1]
(e) sec^-1 \(\frac{1}{x}\) = cos^-1x, x ∈ [-1,1]
(f) cot^-1 \(\frac{1}{x}\) = tan^-1x, x > 0.

2. (a) sin^-1(-x) = – sin^-1x, x ∈ [-1,1]
(b) tan^-1(-x) = – tan^-1x, x ∈ R
(c) cosec^-1(-x) = – cosec^-1x, | x | >1
(d) cos^-1(-x) = π – cos^-1x, x ∈ [-1,1]
(e) sec^-1(-x) = π – sec^-1x, x ∈ R
(f) cot^-1(-x) = π – cot^-1x, x ∈ R

3. (a) sin^-1x + cos^-1x = \(\frac{x}{2}\), x ∈ [-1,1]
(b tan^-1x + cot^-1x = \(\frac{x}{2}\), x ∈ R
(c) cosec^-1x + sec^-1x = \(\frac{x}{2}\), | x | ≥ 1

4. (a) tan^-1x + tan^-1y = tan-1\(\frac{x+y}{1-x y}\), xy < 1
(b) tan^-1x – tan^-1y = tan-1\(\frac{x-y}{1+x y}\), xy > -1
(c) 2 tan^-1x = tan^-1\(\frac{2 x}{1-x^{2}}\), | x | <1
= sin^-1\(\frac{2 x}{1+x^{2}}\), | x | <0
= cos^-1\(\frac{1-x^{2}}{1+x^{2}}\) , x ≥ 0
(d) 2sin^-1x = sin^-1[2x\(\sqrt{1-x^{2}}\)]
(e) 2cos^-1x = cos^-1(2x² – 1)

1. Table

Function	Principal Value Branch
	Domain	Range
(i) y = sin^-1 x	– 1 ≤ x ≤ 1	\(-\frac{\pi}{2}\) ≤ y ≤ \(\frac{\pi}{2}\)
(ii) y = cos^-1 x	– 1 ≤ x ≤ 1	0 ≤ y ≤ π
(iii) y = tan^-1x	– ∞ < x < ∞	\(-\frac{\pi}{2}\) < y < \(\frac{\pi}{2}\)
(iv) y = cot^-1 x	– ∞ < x ≤ -1	0 < y < \(\frac{\pi}{2}\)
(v) y = sec^-1 x	1 ≤ x < ∞ -∞ < x ≤ -1	0 ≤ y < \(\frac{\pi}{2}\) \(\frac{\pi}{2}\)< y < π
(vi) y = cosec^-1 x	1 ≤ x < ∞ -∞ < x ≤ ∞	0 < y ≤ \(\frac{\pi}{2}\) \(-\frac{\pi}{2}\) ≤ y < 0

2. PROPERTIES :

(i) x = sin^-1(sin x) = cos^-1(cos x) = tan^-1 (tan x); etc.
(ii) (a) cosec^-1x = sin^-1 \(\frac { 1 }{ x }\) , x ≥ 1 or x ≤ -1
(b) sec^-1 x = cos^-1\(\frac { 1 }{ x }\) ,x ≥ 1 or x ≤ -1
(c) cot^-1x = tan^-1\(\frac { 1 }{ x }\), x > 0.

(iii) (a) sin^-1 (- x) = – sin^-1x, x ∈ [-1,11
(b) cos^-1 (-x) = π – cos^-1x, x ∈ [- 1, 1]
(c) tan^-1(- x) = – tan^-1 x, x ∈ R
(d) cot^-1(-x) = π – cot^-1 x, x ∈ R
(e) sec^-1 (-x) = π – sec^-1 x, |x| ≥ 1
(f) cosec^-1 (- x) = – cosec^-1 x, |x| ≥ 1

(iv) (a) sin^-1 x + cos^-1 x = \(\frac{\pi}{2}\), x ∈ (-1,1)
(b) tan ^-1 x + cot^-1x = \(\frac{\pi}{2}\) x ∈ R
(c) sec^-1x + cosec^-1= \(\frac{\pi}{2}\), |x| ≥ 1
(d) tan^-1 x + tan^-1 = tan^-1 \(\frac{x+y}{1-x y}\), xy > -1
(e) tan^-1x – tan^-1y = tan^-1 \(\frac{x-y}{1+x y}\), xy > -1
(f) 2tan^-1 x = sin^-1 \(\frac{2 x}{1+x^{2}}\) = cos^-1 \(\frac{2 x}{1-x^{2}}\) = tan^-1

(v) (a) sin ^-1x ± sin^-1 y = sin ^-1 \(\left(x \sqrt{1-y^{2}} \pm y \sqrt{1-x^{2}}\right)\)
(b) cos ^-1x ± cos^-1y = cos ^-1 \(\left(x y \mp \sqrt{1-x^{2}} \sqrt{1-y^{2}}\right)\)