# Inverse Trigonometric Functions — Formulas, Domain, Range

[#Trigonometry](/content/tag/trigonometry/index.html)

TL;DR

The inverse trigonometric functions — arcsin, arccos, arctan, arccsc, arcsec, arccot — undo the standard trig functions. Each takes a ratio and returns an angle. The trick is that sine, cosine, and tangent each map many angles to the same ratio, so their inverses only work on restricted "principal" intervals: arcsin on [−π/2,π/2], arccos on [0,π], arctan on (−π/2,π/2).

## A Function That Had to Pick a Lane

Sine is periodic — it sends infinitely many angles to the same ratio. If you ask "what angle has sine 1/2?" — there's no single answer. To define an inverse function that gives back exactly one angle, mathematicians had to _pick a lane_: a single interval on which sine is one-to-one. That interval is the **principal branch**, and it's the reason every inverse trig function has a restricted range.

## What Are Inverse Trigonometric Functions?

**Inverse trigonometric functions** are the functions that "undo" the basic trig functions. Each takes a real number (a trig ratio) as input and returns an angle as output:

arcsin⁡x=θ⟺sin⁡θ=x;(with θ in principal range)

The six functions are written either with "arc" notation (arcsinx, arccosx, etc.) or with the "−1" superscript (sin⁡−1x, cos⁡−1x, etc.). They are functions, so each input gives exactly one output — which is the whole point of restricting the range.

## The Six Inverse Trigonometric Formulas

| Inverse function | Notation | Domain         | Range (principal)          |
|------------------|----------|----------------|-----------------------------|
| Arcsine          | arcsinx  | [-1, 1]        | [-π/2, π/2]                |
| Arccosine        | arccosx  | [-1, 1]        | [0, π]                     |
| Arctangent       | arctanx  | (−∞, ∞)         | (−π/2, π/2)                |
| Arccosecant      | arccscx  | (−∞, −1] ∪ [1, ∞) | [−π/2, 0) ∪ (0, π/2]      |
| Arcsecant        | arcsecx  | (−∞, −1] ∪ [1, ∞) | [0, π/2) ∪ (π/2, π]      |
| Arccotangent     | arccotx  | (−∞, ∞)         | (0, π)                     |

**The pattern.** Domain of each inverse function = range of the original function. Range of each inverse function = the principal interval where the original is one-to-one. The "−1" in sin⁡−1x is _not_ a reciprocal — it's the inverse-function label.

## Why the Range Is Restricted

Take arcsine. Sine, viewed as a function of all real numbers, is _not_ one-to-one — infinitely many angles share the same sine value. To define an inverse, we restrict sine to the interval [−π/2,π/2], on which sine _is_ one-to-one and continuous. The inverse of that restricted sine is what we call arcsine, with range [−π/2,π/2].

Similarly:

- Cosine is restricted to [0,π] — that's arccos's range.
- Tangent is restricted to (−π/2,π/2) — that's arctan's range.

These choices aren't arbitrary; they're the largest connected intervals containing zero on which each function is monotonic.

## Three Worked Examples — Quick, Standard, Stretch

### Quick

**Find arcsin(1/2).**  We want the angle θ∈[−π/2,π/2] with sin⁡θ=1/2. From the unit circle, sin(π/6)=1/2 and π/6 is in the principal range.

arcsin(1/2)=π/6=30°

### Wrong Path First — Then the Right One — Standard Example

**Find arcsin(sin(2π/3)).**  _The wrong path._ A student writes: "Arcsine undoes sine, so arcsin(sin(2π/3))=2π/3." Done.

That answer is wrong, even though it looks like an "inverse undoing the original" must give back the input.

**Sanity check.** Arcsine returns values only in [−π/2,π/2]. The angle 2π/3 is _outside_ that range (2π/3≈120° — past π/2=90°). So arcsin(sin(2π/3)) cannot possibly equal 2π/3.

_The correct path._ Compute sin(2π/3) first — using the reference angle, sin(2π/3)=sin(π−2π/3)=sin(π/3)=√3/2 (positive because 2π/3 is in quadrant II).

Then arcsin(√3/2)=π/3 — the angle in [−π/2,π/2] whose sine is √3/2.

arcsin(sin(2π/3))=π/3, not 2π/3.

### Stretch

**Find the exact value of sin(2arctan(3/4)).**

Let θ=arctan(3/4), so tan⁡θ=3/4 and θ∈(−π/2,π/2). Imagine a right triangle with opposite side 3, adjacent side 4 — then hypotenuse is √(3²+4²)=5.

So sin⁡θ=3/5 and cos⁡θ=4/5.

Using the double-angle formula: sin(2θ)=2sin(θ)cos(θ)=2⋅(3/5)⋅(4/5)=24/25.

sin(2arctan(3/4))=24/25.

The trick — turning an inverse-trig expression into a right triangle — works for any composition like sin(arctan(x)), cos(arcsin(x)), etc. Worth memorising.

## Graphs of Inverse Trigonometric Functions

Each inverse-function graph is the reflection of the corresponding restricted forward function across the line y=x.

- **Arcsine.** S-shaped curve from (−1,−π/2) to (1,π/2), passing through the origin.
- **Arccosine.** Mirror image of arcsine — from (−1,π) to (1,0), passing through (0,π/2). Strictly decreasing.
- **Arctangent.** Smooth S-curve passing through the origin, with horizontal asymptotes at ±π/2.

## Derivatives of Inverse Trigonometric Functions

These are the most-used inverse-trig formulas in calculus:

\[ \frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}} \qquad \frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}} \]
\[ \frac{d}{dx}\arctan x = \frac{1}{1+x^2} \qquad \frac{d}{dx}\arccot x = -\frac{1}{1+x^2} \]
\[ \frac{d}{dx}\arcsec x = \frac{1}{|x|\sqrt{x^2-1}} \qquad \frac{d}{dx}\arccsc x = -\frac{1}{|x|\sqrt{x^2-1}} \]

**The pattern.** Each "co-" inverse has the negative of its partner's derivative — because arccos x + arcsin x = π/2 for all x∈[−1,1]. Differentiate both sides and the negative sign drops out.

## Identities Worth Knowing

\[ \, arcsin x + arccos x = \frac{\pi}{2} \quad \arctan x + \arccot x = \frac{\pi}{2} \]  
\[ \, arcsin(-x) = -arcsin x \quad arctan(-x) = -arctan x \]  
\[ \, arccos(-x) = \pi - arccos x \quad arctan x + arctan(1/x) = \pm\frac{\pi}{2} \]

The first row is the most useful — it lets you convert between any two complementary inverses in one step.

## Why Inverse Trig Functions Show Up Everywhere

The forward trig functions answer "given an angle, what's the ratio?" The inverse functions answer the more practical question: "given a measured ratio, what angle produced it?"

- **Surveying.** A surveyor measures horizontal and vertical distances and needs the angle — arctan(rise/run). 
- **Robotics and inverse kinematics.** When a robot arm needs to reach a target position, the controller solves for the joint angles by inverting the forward-kinematics equations.
- **GPS and navigation.** Latitude is recovered from celestial measurements using arcsin. Bearing angles use arctan. 
- **Computer vision.** The orientation of a detected line in an image is arctan(slope). 
- **Physics — refraction.** Snell's law gives the angle of refraction as arcsin(n₁sinθ₁/n₂).

The forward trig functions matter; the inverse ones matter more in any setting where the measurement comes first and the angle has to be recovered.

## The Mathematicians Who Shaped Inverse Trigonometric Functions

Inverse trig functions developed alongside their forward counterparts, but the principal-branch convention came surprisingly late.

- **Daniel Bernoulli** (1700–1782, Swiss) — first used the notation A.sin A for arcsine in 1729. 
- **Leonhard Euler** (1707–1783, Swiss) — formalised inverse trig functions in 1748. 
- **Augustin-Louis Cauchy** (1789–1857, French) — formalised the principal-branch concept in the 1820s.

**Why it matters:** every time a calculator returns arcsin(0.5)=π/6 instead of, say, 5π/6 — it's executing Cauchy's principal-branch convention from 1821.

## Key Takeaways

- The six **inverse trigonometric functions** undo the six standard trig functions, returning an angle from a ratio.
- Each has a restricted "principal" range because the original function is periodic and not one-to-one.
- The biggest exam slip is assuming arcsin(sin(x))=x for every x — it only holds when x is already in [−π/2,π/2].
- Inverse trig functions power surveying, robotics, GPS, computer vision, and refraction calculations.

## Try It Yourself — Three Problems

1. Compute arccos(cos(7π/4)). (Check whether 7π/4 is in [0,π]).
2. Evaluate sin(arccos(0.6)) by drawing a right triangle.
3. Differentiate f(x)=arctan(2x) using the chain rule.