Inverse Trigonometric Functions
Arcsine, Arccosine, Arctangent & Arccotangent — Definitions, Properties, Compositions, and Applications
1. The Invertibility Problem
Why Can't We Just "Invert" Sine, Cosine, or Tangent?
Recall from our earlier study of inverse functions that a function $ f $ has an inverse $ f^{-1} $ if and only if $ f $ is one-to-one (injective). A function is one-to-one when every horizontal line crosses its graph at most once.
The Solution: Domain Restriction
We restrict each trigonometric function to a carefully chosen interval where it is:
- Strictly monotonic (either always increasing or always decreasing) — guaranteeing injectivity.
- Surjective onto its full range — so every output value is still attainable.
Together, these two conditions make the restricted function a bijection, and a bijection always has an inverse.
The specific intervals are chosen to include the first-quadrant angles $ (0, \pi/2) $ and to be as large as possible while maintaining monotonicity. This is a natural and standard convention used across mathematics.
2. Definitions — Domains and Ranges
Arcsine: $ y = \arcsin x $
Definition: $ y = \arcsin x $ means $ \sin y = x $ and $ y \in \left[-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right] $.
In words: the unique angle between $ -90° $ and $ 90° $ whose sine equals $ x $.
| Domain | $ [-1,\, 1] $ |
|---|---|
| Range | $ \left[-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right] $ |
Why this interval? $ \sin x $ is strictly increasing on $ [-\pi/2,\, \pi/2] $ and takes every value from $ -1 $ to $ 1 $.
Arccosine: $ y = \arccos x $
Definition: $ y = \arccos x $ means $ \cos y = x $ and $ y \in [0,\, \pi] $.
In words: the unique angle between $ 0° $ and $ 180° $ whose cosine equals $ x $.
| Domain | $ [-1,\, 1] $ |
|---|---|
| Range | $ [0,\, \pi] $ |
Why this interval? $ \cos x $ is strictly decreasing on $ [0,\, \pi] $ and takes every value from $ -1 $ to $ 1 $.
Arctangent: $ y = \arctan x $
Definition: $ y = \arctan x $ means $ \tan y = x $ and $ y \in \left(-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right) $.
In words: the unique angle strictly between $ -90° $ and $ 90° $ whose tangent equals $ x $.
| Domain | $ (-\infty,\, +\infty) = \mathbb{R} $ |
|---|---|
| Range | $ \left(-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right) $ |
Why this interval? $ \tan x $ is strictly increasing on $ (-\pi/2,\, \pi/2) $ and takes every real value. The endpoints are excluded because $ \tan x $ has vertical asymptotes there.
Arccotangent: $ y = \text{arccot}\, x $
Definition: $ y = \text{arccot}\, x $ means $ \cot y = x $ and $ y \in (0,\, \pi) $.
In words: the unique angle strictly between $ 0° $ and $ 180° $ whose cotangent equals $ x $.
| Domain | $ (-\infty,\, +\infty) = \mathbb{R} $ |
|---|---|
| Range | $ (0,\, \pi) $ |
Why this interval? $ \cot x $ is strictly decreasing on $ (0,\, \pi) $ and takes every real value.
Summary Table
| Function | Notation | Domain | Range | Monotonicity |
|---|---|---|---|---|
| Arcsine | $ \arcsin x $ | $ [-1, 1] $ | $ [-\pi/2,\, \pi/2] $ | Increasing |
| Arccosine | $ \arccos x $ | $ [-1, 1] $ | $ [0,\, \pi] $ | Decreasing |
| Arctangent | $ \arctan x $ | $ \mathbb{R} $ | $ (-\pi/2,\, \pi/2) $ | Increasing |
| Arccotangent | $ \text{arccot}\, x $ | $ \mathbb{R} $ | $ (0,\, \pi) $ | Decreasing |