Introduction

2.1 Introduction

The Problem with Trigonometric Inverses

From Chapter 1, the inverse of a function $f$, written $f^{-1}$, exists only when $f$ is both one‑one (injective) and onto (surjective). The trigonometric functions $\sin x$, $\cos x$, $\tan x$, $\csc x$, $\sec x$, $\cot x$ fail one or both conditions over their natural domains and ranges. Consider $\sin x$: it repeats every $2\pi$ (so it is not one‑one) and its range is only $[-1, 1]$, not all of $\mathbb{R}$ (so it is not onto). Consequently their inverses do not exist over these natural domains and ranges.

The Solution: Restricting Domains and Ranges

To obtain invertible trigonometric functions we must restrict their domains and ranges to intervals where they become one‑one and onto. By carefully choosing such intervals we can define $\sin^{-1} x$, $\cos^{-1} x$, $\tan^{-1} x$, $\csc^{-1} x$, $\sec^{-1} x$, $\cot^{-1} x$. In this chapter we will:

• specify the exact restricted domain and range that make each inverse exist;
• examine the graphs of these inverse functions and discuss their monotonicity, symmetry, and asymptotes;
• prove elementary identities, each derived step by step with full justification, such as:
  • $\sin^{-1}(-x) = -\sin^{-1} x$
  • $\cos^{-1}(-x) = \pi - \cos^{-1} x$
  • $\tan^{-1}(-x) = -\tan^{-1} x$
  • $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$
  • $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$
  • $\csc^{-1} x + \sec^{-1} x = \frac{\pi}{2}$

Why Inverse Trigonometric Functions Matter

These functions play a vital role in calculus, where they appear as antiderivatives of many algebraic expressions — for example $\int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1} x + C$. They are also used extensively in science and engineering: in physics for analysing waves and oscillations, in signal processing, and in geometry for solving triangles.