CBSE Class 12 Mathematics Inverse Trigonometric Function Notes Set 01

BASIC CONCEPTS

1. Definition: If \( f: X \rightarrow Y \) is one-one onto (bijective) function, then there exists a unique function \( f^{-1}: Y \rightarrow X \) which assigns each element \( y \in Y \) to a unique element \( x \in X \) such that \( f(x) = y \) and is called inverse function of \( f \).
i.e., \( f^{-1}(y) = x \Leftrightarrow f(x) = y, \ x \in X \text{ and } y \in Y \)

Principal value branches: Since trigonometric functions being periodic are in general not bijective (one-one onto) and thus for existence of inverse of trigonometric function we restrict their domain and co-domain to make it bijective. This restriction of domain and range gives principal value branch of inverse trigonometric function which are as follows:

The value of an inverse trigonometric function which lies in its principal value branch is called the principal value of that inverse trigonometric function.

2. Principal and general values:

• (a) If \( \sin \theta = \sin \alpha \) then its principal value is \( \theta = \alpha, -\frac{\pi}{2} < \alpha < \frac{\pi}{2} \) and its general value is \( \theta = n\pi + (-1)^n \alpha, n \in Z \)
• (b) If \( \cos \theta = \cos \alpha \) then its principal value is \( \theta = \alpha, 0 < \alpha < \pi \) and its general value is \( \theta = 2n\pi \pm \alpha, n \in Z \)
• (c) If \( \tan \theta = \tan \alpha \) then its principal value is \( \theta = \alpha, -\frac{\pi}{2} < \alpha < \frac{\pi}{2} \) and its general value is \( \theta = n\pi + \alpha, n \in Z \)

Properties of Inverse Trigonometric Functions

1.
(i) \( \sin^{-1}(\sin \theta) = \theta \), for all \( \theta \in [-\pi/2, \pi/2] \)
(ii) \( \cos^{-1}(\cos \theta) = \theta \), for all \( \theta \in [0, \pi] \)
(iii) \( \tan^{-1}(\tan \theta) = \theta \), for all \( \theta \in (-\pi/2, \pi/2) \)
(iv) \( \text{cosec}^{-1}(\text{cosec } \theta) = \theta \), for all \( \theta \in [-\pi/2, \pi/2], \theta \neq 0 \)
(v) \( \sec^{-1}(\sec \theta) = \theta \), for all \( \theta \in [0, \pi], \theta \neq \pi/2 \)
(vi) \( \cot^{-1}(\cot \theta) = \theta \), for all \( \theta \in (0, \pi) \)

2.
(i) \( \sin(\sin^{-1} x) = x \), for all \( x \in [-1, 1] \)
(ii) \( \cos(\cos^{-1} x) = x \), for all \( x \in [-1, 1] \)
(iii) \( \tan(\tan^{-1} x) = x \), for all \( x \in R \)
(iv) \( \text{cosec}(\text{cosec}^{-1} x) = x \), for all \( x \in (-\infty, -1] \cup [1, \infty) \)
(v) \( \sec(\sec^{-1} x) = x \), for all \( x \in (-\infty, -1] \cup [1, \infty) \)
(vi) \( \cot(\cot^{-1} x) = x \), for all \( x \in R \)

3.
(i) \( \sin^{-1}\left(\frac{1}{x}\right) = \text{cosec}^{-1} x \), for all \( x \in (-\infty, -1] \cup [1, \infty) \)
(ii) \( \cos^{-1}\left(\frac{1}{x}\right) = \sec^{-1} x \), for all \( x \in (-\infty, -1] \cup [1, \infty) \)
(iii) \( \tan^{-1}\left(\frac{1}{x}\right) = \begin{cases} \cot^{-1} x, & \text{for all } x > 0 \\ -\pi + \cot^{-1} x, & \text{for all } x < 0 \end{cases} \)

4.
(i) \( \sin^{-1}(-x) = -\sin^{-1} x \), for all \( x \in [-1, 1] \)
(ii) \( \cos^{-1}(-x) = \pi - \cos^{-1} x \), for all \( x \in [-1, 1] \)
(iii) \( \tan^{-1}(-x) = -\tan^{-1} x \), for all \( x \in R \)
(iv) \( \text{cosec}^{-1}(-x) = -\text{cosec}^{-1} x \), for all \( x \in (-\infty, -1] \cup [1, \infty) \)
(v) \( \sec^{-1}(-x) = \pi - \sec^{-1} x \), for all \( x \in (-\infty, -1] \cup [1, \infty) \)
(vi) \( \cot^{-1}(-x) = \pi - \cot^{-1} x \), for all \( x \in R \)

5.
(i) \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \), for all \( x \in [-1, 1] \)
(ii) \( \tan^{-1} x + \cot^{-1} x = \frac{\pi}{2} \), for all \( x \in R \)
(iii) \( \sec^{-1} x + \text{cosec}^{-1} x = \frac{\pi}{2} \), for all \( x \in (-\infty, -1] \cup [1, \infty) \)

6.
(i) \( \sin^{-1} x + \sin^{-1} y = \sin^{-1}\{x\sqrt{1 - y^2} + y\sqrt{1 - x^2}\} \), if \( -1 \le x, y \le 1 \) and \( x^2 + y^2 \le 1 \) or if \( xy < 0 \) and \( x^2 + y^2 > 1 \)
(ii) \( \sin^{-1} x - \sin^{-1} y = \sin^{-1}\{x\sqrt{1 - y^2} - y\sqrt{1 - x^2}\} \), if \( -1 \le x, y \le 1 \) and \( x^2 + y^2 \le 1 \) or if \( xy > 0 \) and \( x^2 + y^2 > 1 \)

7.
(i) \( \cos^{-1} x + \cos^{-1} y = \cos^{-1}\{xy - \sqrt{1 - x^2}\sqrt{1 - y^2}\} \), if \( -1 \le x, y \le 1 \) and \( x + y \ge 0 \)
(ii) \( \cos^{-1} x - \cos^{-1} y = \cos^{-1}\{xy + \sqrt{1 - x^2}\sqrt{1 - y^2}\} \), if \( -1 \le x, y \le 1 \) and \( x \le y \)

8.
(i) \( \tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x + y}{1 - xy} \right) \), if \( xy < 1 \)
(ii) \( \tan^{-1} x - \tan^{-1} y = \tan^{-1} \left( \frac{x - y}{1 + xy} \right) \), if \( xy > -1 \)

9.
(i) \( 2\sin^{-1} x = \sin^{-1} (2x\sqrt{1 - x^2}) \), if \( -\frac{1}{\sqrt{2}} \le x \le \frac{1}{\sqrt{2}} \)
(ii) \( 2\cos^{-1} x = \cos^{-1} (2x^2 - 1) \), if \( 0 \le x \le 1 \)
(iii) \( 2\tan^{-1} x = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \), if \( -1 < x < 1 \)

10.
(i) \( 3\sin^{-1} x = \sin^{-1} (3x - 4x^3) \), if \( -\frac{1}{2} \le x \le \frac{1}{2} \)
(ii) \( 3\cos^{-1} x = \cos^{-1} (4x^3 - 3x) \), if \( \frac{1}{2} \le x \le 1 \)
(iii) \( 3\tan^{-1} x = \tan^{-1} \left( \frac{3x - x^3}{1 - 3x^2} \right) \), if \( -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}} \)

11.
(i) \( 2\tan^{-1} x = \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), if \( -1 \le x \le 1 \)
(ii) \( 2\tan^{-1} x = \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \), if \( 0 \le x < \infty \)

12.
(i) \( \sin^{-1} x = \cos^{-1}(\sqrt{1 - x^2}) = \tan^{-1}\left(\frac{x}{\sqrt{1 - x^2}}\right) = \cot^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right) = \sec^{-1}\left(\frac{1}{\sqrt{1 - x^2}}\right) = \text{cosec}^{-1}\left(\frac{1}{x}\right) \)
(ii) \( \cos^{-1} x = \sin^{-1}(\sqrt{1 - x^2}) = \tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right) = \cot^{-1}\left(\frac{x}{\sqrt{1 - x^2}}\right) = \sec^{-1}\left(\frac{1}{x}\right) = \text{cosec}^{-1}\left(\frac{1}{\sqrt{1 - x^2}}\right) \)
(iii) \( \tan^{-1} x = \sin^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right) = \cos^{-1}\left(\frac{1}{\sqrt{1 + x^2}}\right) = \cot^{-1}\left(\frac{1}{x}\right) = \sec^{-1}(\sqrt{1 + x^2}) = \text{cosec}^{-1}\left(\frac{\sqrt{1 + x^2}}{x}\right) \)

Important substitution to simplify trigonometrical expressions involving inverse trigonometrical functions.

Important Facts:

• (i) If no branch of an inverse trigonometric function is mentioned, we mean the principal value branch of that function.

• (ii) \( \sin^{-1} x \neq \frac{1}{\sin x} \) or \( (\sin x)^{-1} \) and same holds true for other trigonometric functions also.
• (iii) If \( \sin^{-1} x = y \) then \( x \) and \( y \) are the elements of domain and range of principal value branch of \( \sin^{-1} \) respectively.
  i.e., \( x \in [-1, 1] \) and \( y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)

Similar fact is also applicable for other inverse trigonometric functions.