To understand principal values of inverse trigonometric functions, let’s break it down step by step:
1. Why Do We Need Inverse Trig Functions?
Trigonometric functions (like \( \sin \theta \), \( \cos \theta \), \( \tan \theta \)) take an angle and give a number. Inverse trig functions (like \(\sin^{-1} x\), \(\cos^{-1} x \), \(\tan^{-1} x \)) do the opposite: they take a number and return an angle.
2. The Problem: Periodicity
Trig functions are periodic, meaning they repeat their values. For example:
- \( \sin 30^\circ = \sin 150^\circ = 0.5 \)
- \( \cos 0^\circ = \cos 360^\circ = 1 \)
This means there are infinitely many angles that give the same trig value. But a function can only have one output for each input. So, how do we define inverse trig functions?
3. Solution: Restrict the Domain (Principal Values)
To make inverse trig functions work, we restrict their range (the angles they can output) to a specific interval called the principal value. This ensures each input gives exactly one angle.
Principal Value Ranges:
| Function | Principal Value Range | Why? |
|---|---|---|
| \( \sin^{-1}x \) | \([- \frac{\pi}{2}, \frac{\pi}{2}] \) | Covers all outputs from \(-1 \) to \(1 \) and is centred around 0. |
| \( \cos^{-1}x \) | \([0, \pi] \) | Covers all outputs from \(-1 \) to \(1 \) and includes all quadrants. |
| \( \tan^{-1}x \) | \( (- \frac{\pi}{2}, \frac{\pi}{2}) \) | Avoids vertical asymptotes and covers all real numbers. |
4. Key Takeaways
- Principal values are the “main” angles returned by inverse trig functions.
- Calculators use these ranges to give a single answer (e.g., typing \( \sin^{-1}(0.5) \) gives \( \frac{\pi}{6} \)).