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# Arctan — Formula, Graph, Identities, Domain and Range

## TL;DR
Arctan is the inverse tangent — it takes a real number and returns the angle whose tangent is that number. Its domain is every real number (−∞,∞), its range is the open interval (−π/2, π/2), and its graph is a smooth S-curve with horizontal asymptotes at y=±π/2.

## A Radar Operator's Quiet Calculation
A radar dish picks up an aircraft 12 km out and 3 km up — what's the bearing angle? The answer is arctan(3/12)≈14°, computed roughly two thousand times per second by air-traffic systems around the world. Tangent goes one way: angle in, ratio out. Arctan goes the other way: ratio in, angle out.

## What Is Arctan?
**Arctan** is the inverse of the tangent function. If tan θ=x, then arctan x=θ — provided θ is restricted to a range where tangent is one-to-one, which we'll cover in a moment. The function is also written tan⁻¹x (the "−1" is a label, not a reciprocal — tan⁻¹x≠1/tan x).

Arctan takes any real number and returns an angle. That's the whole job.

## The Arctan Formula
The defining relationship is the simplest possible:

y=arctan x ⟺ tan y=x; and; y∈(−π/2,π/2)

In a right-triangle context, if the side opposite an acute angle θ is p and the adjacent side is b, then:

θ=arctan(p/b)

Five quick values to memorise — they appear constantly:

| xxx | arctan x (radians) | arctan x (degrees) |
| --- | --- | --- |
| 0 | 0 | 0° |
| 1 | π/4 | 45° |
| √3 | π/3 | 60° |
| 1/√3 | π/6 | 30° |
| −1 | −π/4 | −45° |

## Domain and Range of Arctan
| Property | Value |
| --- | --- |
| **Domain** | (−∞,∞) — every real number |
| **Range** | (−π/2,π/2) — open interval, asymptotes never reached |
| **Asymptotes** | y=π/2 (right), y=−π/2 (left) |
| **Continuity** | Continuous everywhere |
| **Symmetry** | Odd function — arctan(−x)=−arctan(x) |
| **Monotonicity** | Strictly increasing |

The domain is wide open because tangent's range is wide open — tangent already hits every real number on (−π/2,π/2), so inverting it gives a function defined on the same set.

The range is restricted on purpose. Tangent is periodic with period π — without a restriction, infinitely many angles would map to the same tangent value, and the inverse wouldn't be a function. Mathematicians chose (−π/2,π/2) as the **principal branch** because it's the largest interval containing 0 on which tangent is one-to-one and continuous.

## The Arctan Graph
Picture a sine wave laid sideways and pulled flat at the top and bottom. That's the shape — a soft S-curve that approaches but never touches two horizontal lines.

**Three features worth pinning down:**

- **Passes through the origin.** arctan(0)=0.
- **Horizontal asymptotes.** As x→∞, arctan x→π/2. As x→−∞, arctan x→−π/2.
- **Steepest at x=0.** The slope at the origin is exactly 1 — confirmed by the derivative below.

## Arctan Identities
Identities that come up in problems and proofs:

arctan(−x)=−arctan(x)

arctan(x) + arctan(1/x) = π/2 for x>0

arctan(x) + arctan(1/x) = −π/2 for x<0

arctan(x) + arctan(y) = arctan((x+y)/(1−xy)) + kπ

where k=0 if xy<1, k=1 if xy>1 and both are positive, and k=−1 if xy>1 and both are negative. **The k correction is a common mistake.**

### Arctan derivative and integral
\frac{d}{dx}arctan(x)=\frac{1}{1+x^2}

∫arctan(x) dx = x arctan(x) − \frac{1}{2}ln(1+x^2) + C

The derivative is bounded (≤1, equal to 1 at x=0) — which matches the flattening tails of the graph.

## Three Worked Examples — Quick, Standard, Stretch
### Quick
**Find arctan(1).**
We're asking: which angle in (−π/2,π/2) has tangent equal to 1? From the 45-45-90 triangle, tan(π/4)=1. So:

arctan(1)=π/4=45°.

### The Tempting Shortcut That Doesn't Work — A Worked Standard Example
**Find arctan(3)+arctan(1).**
_The tempting shortcut._ A student sees the addition formula and writes:

arctan(3)+arctan(1)=arctan((3+1)/(1−3⋅1))

Rationalising:

(3+1)(1+3)/(1−3)=(3+1)^2/(1-3) = (4+2√3)/-2.

So the student gets arctan(−(2+√3))=−arctan(2+√3)

**Sanity check.** arctan(3)=60° and arctan(1)=45° — both clearly positive. Their sum is 105°.

The correction. Here, x=3, y=1, and xy=3>1 so the k=1 branch correction applies:

arctan(3)+arctan(1)=−arctan(2+√3)+π.

### Stretch
**A ladder leans against a wall. Its foot is 4 m from the wall, its top is 9 m up the wall. What angle does the ladder make with the ground?**

θ=arctan(9/4)≈1.1526 rad≈66.04°

The ladder makes a 66-degree angle with the ground.

## Why Engineers, Pilots, and Coders Still Use Arctan
Arctan is the angle-recovery function.

- **GPS and navigation.** Every bearing computation — is arctan(y/x).
- **Robotics.** Inverse kinematics — figuring out joint angles that put an end-effector at a target position.
- **Graphics.** Rotating a sprite to point at the mouse cursor is arctan(cursor.y - sprite.y, cursor.x - sprite.x).
- **Signal processing.** The phase of a complex number is arctan(b/a).

## Slip-Ups That Cost Marks on Arctan
### **Mistake 1:** Treating tan⁻¹x as 1/tan x
### **Mistake 2:** Ignoring the principal range when solving tan θ=k
### **Mistake 3:** Forgetting the k correction in arctan(x)+arctan(y)
### **Mistake 4:** Using arctan(y/x) when you should be using atan2(y, x)

## Key Takeaways
- **Arctan** is the inverse tangent — input is any real number, output is an angle in (−π/2,π/2).
- The derivative is \dfrac{1}{1+x^2} — bounded above by 1.
- The biggest exam-day slip is treating tan⁻¹x as 1/tan x — they are not the same.
- For 2D vector angles, use atan2(y, x), not arctan(y/x).