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 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 [\(\frac{\pi}{2}\), \(\frac{3\pi}{2}\)] …….
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.
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.
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}……..
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] – {\(\frac{3 \pi}{2}\)}, ……………
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.
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:
→ 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 \(\left(x y \mp \sqrt{1-x^{2}} \sqrt{1-y^{2}}\right)\)