Class 12 - Inverse Trigonometric Functions

1. Introduction

• Definition: Recall that a function $f$ has an inverse $f^{-1}$ if and only if it is one-one and onto (bijective).

• The Problem with Trig Functions: Trigonometric functions are periodic and thus not one-one over their entire natural domains.

• The Solution: By restricting the domains of trigonometric functions, we can make them one-one and onto, allowing us to define their inverses.

• Applications: Used extensively in Calculus (integration), science, and engineering.

2. Basic Concepts

• Inversion of Functions:

  • If $y = \sin x$, then $x = \sin^{-1} y$ (provided $x$ is in the restricted domain).

  • Note: $\sin^{-1} x$ is an angle, not to be confused with $(\sin x)^{-1}$ or $\frac{1}{\sin x}$.

• Principal Value Branch: The specific range of an inverse trigonometric function where it is defined as a unique function.

• Domain and Range Table: $y = \sin^{-1} x$ | $[-1, 1]$ | $[-\frac{\pi}{2}, \frac{\pi}{2}]$ | $y = \cos^{-1} x$ | $[-1, 1]$ | $[0, \pi]$ | $y = \tan^{-1} x$ | $\mathbb{R}$ | $(-\frac{\pi}{2}, \frac{\pi}{2})$ | $y = \text{cosec}^{-1} x$ | $\mathbb{R} - (-1, 1)$ | $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$ | $y = \sec^{-1} x$ | $\mathbb{R} - (-1, 1)$ | $[0, \pi] - \{\frac{\pi}{2}\}$ | $y = \cot^{-1} x$ | $\mathbb{R}$ | $(0, \pi)$ |

• Principal Value: The value of an inverse trigonometric function that lies within its principal value branch.

3. Properties of Inverse Trigonometric Functions

• Self-Inverse Properties:

  • $\sin(\sin^{-1} x) = x, \text{ for } x \in [-1, 1]$

  • $\sin^{-1}(\sin x) = x, \text{ for } x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$

• Reciprocal Identities:

  • $\sin^{-1}(\frac{1}{x}) = \text{cosec}^{-1} x, x \geq 1$ or $x \leq -1$

  • $\cos^{-1}(\frac{1}{x}) = \sec^{-1} x, x \geq 1$ or $x \leq -1$

  • $\tan^{-1}(\frac{1}{x}) = \cot^{-1} x, x > 0$

• Negative Argument Identities:

  • $\sin^{-1}(-x) = -\sin^{-1} x$

  • $\tan^{-1}(-x) = -\tan^{-1} x$

  • $\cos^{-1}(-x) = \pi - \cos^{-1} x$

• Complementary Angle Identities:

  • $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, x \in [-1, 1]$

  • $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}, x \in \mathbb{R}$

  • $\text{cosec}^{-1} x + \sec^{-1} x = \frac{\pi}{2}, |x| \geq 1$

• Sum and Difference of $\tan^{-1}$:

  • \tan^{-1} x + \tan^{-1} y = \tan^{-1}\left(\frac{x+y}{1-xy}\right), xy < 1

  • $\tan^{-1} x - \tan^{-1} y = \tan^{-1}\left(\frac{x-y}{1+xy}\right), xy > -1$

• Double Angle Conversions:

  • $2\tan^{-1} x = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \tan^{-1}\left(\frac{2x}{1-x^2}\right)$

4. Graphical Representations

• Understanding the visual behavior of inverse functions (e.g., how the graph of $y = \sin^{-1} x$ is the reflection of $y = \sin x$ over the line $y = x$ within the restricted interval).

5. Historical Note

• The study of trigonometry originated in India with ancient mathematicians like Aryabhata (476 A.D.), Brahmagupta, and Bhaskara I.

Quick Summary for Revision

1. Always check the range: When calculating $\sin^{-1}(x)$, ensure your answer is within $[-\pi/2, \pi/2]$.

2. Conversion: You can convert any inverse trig function to another using right-angled triangle properties (Pythagoras theorem).

3. Simplification: Use the $\tan^{-1}$ properties for most proof-based problems as they are often the most straightforward to apply.