Arctan 2 — Value, Radians, Degrees, Worked Examples

Trigonometry

TL;DR

Arctan 2 — the angle whose tangent equals 2 — is approximately 1.1071 radians or 63.435°. This article covers the exact-value status (irrational, non-terminating), three computation methods (calculator, series expansion, right-triangle reading), the unit-circle position, the common composition mistakes, and a quick reference for arctan at nearby integer inputs.

Quick Answer:
Result: arctan(2)≈1.1071487 radians ≈63.434965°
Notation: arctan(2)=tan⁡−1(2) — read as "the angle in (−π/2,π/2) whose tangent is 2." NOT to be confused with 1/tan(2), which is the cotangent of 2 radians.
Method shown: Calculator (primary), Madhava–Gregory series (numerical), right-triangle reading (geometric).
Approximate value (irrational): 1.1071487177940904 rad / 63.43494882292201°
Exact form: no closed-form expression in elementary constants — arctan(2) is not a rational multiple of π and is not constructible from radicals. The value is irrational and transcendental.

Quick Reference — Arctan at Nearby Integer Inputs

The arctan function approaches π/2 (90°) as x→+∞ — it never reaches it. So arctan(2)≈63.4° sits roughly two-thirds of the way from arctan(1)=45° to the asymptote.

Where Arctan 2 Shows Up

The angle arctan(2) shows up wherever a slope of 200% needs to be converted into an inclination angle. The angle a road grade of 200% makes with the horizontal is arctan(2)≈63.4°. The value appears in robotics for two-link arm geometry and in 3D graphics where the elevation angle of a camera ray with a horizontal-to-vertical ratio of 1:2 is exactly arctan(2).

What Is Arctan 2? — The Definition Behind the Number

The arctangent of a real number x is the unique angle θ in the open interval (−π/2,π/2) such that tan(θ)=x. For x=2:

arctan(2)=θ⟺tan(θ)=2; with θ∈(−π/2,π/2).

Because tangent is strictly increasing on this interval and runs from −∞ to +∞, exactly one θ satisfies the equation. That θ is arctan(2)≈1.1071 rad.
In degree-mode, the interval is (−90°,90°) and the answer is ≈63.435°.

Why Arctan 2 Has No Clean Exact Form

The special-angle table — 0°,30°,45°,60°,90° — covers angles whose tangents are 0,1/√3,1,√3,undef. The value 2 is not on that list, and it is not on the wider list of constructible values either.

The number arctan(2) is in fact a transcendental number (a consequence of the Lindemann–Weierstrass theorem). It cannot be written as a finite combination of square roots, cube roots, or other algebraic expressions. The best you can do is compute as many decimals as you need.

Three Ways to Compute Arctan 2

Method 1: Calculator

Switch the calculator to radian mode, press the tan⁡−1 or arctan key, and enter 2. The display reads 1.1071487… rad.
Switch to degree mode and the same calculation reads 63.434965…°.

Final answer: arctan(2)≈1.1071 rad ≈63.435°.

Method 2: Madhava–Gregory series

The Madhava–Gregory series for arctangent is:

arctan(x)=x−x³/3+x⁵/5−x⁷/7+⋯
This converges only for ∣x∣≤1. Use the reflection identity instead:

arctan(2)=π/2−arctan(1/2).

Now apply the series to arctan(1/2) where it converges quickly:

arctan(1/2)≈0.4636.

Then arctan(2)≈π/2−0.4636=1.1072 rad.

In degrees: 90°−26.565°=63.435°.

Final answer: arctan(2)≈1.1072 rad ≈63.435° (matches Method 1).

Method 3: Right-triangle reading

Build a right triangle with legs of length 1 and 2. The hypotenuse is √5. The angle opposite the leg of length 2 has:

tan(θ)=opposite/adjacent=2/1=2.

That angle is arctan(2). Reading values from the triangle:

sin(arctan(2))=2/√5,
cos(arctan(2))=1/√5.

Final answer: arctan(2) is the angle of the diagonal of a 1×2 rectangle measured off the long side — ≈1.1071 rad ≈63.435°.

Common Mistakes With Arctan 2

1. Reading "tan⁡−12" as "1/tan(2)"

Where it slips in: A student sees tan⁡−1(2) on a calculator and types 1÷tan(2) instead.

Don't do this: Treat the superscript −1 as a reciprocal exponent. On trig functions, tan⁡−1 is the inverse function — not the reciprocal.

The correct way: tan⁡−1(2)=arctan(2)≈1.1071 rad.

2. Computing in degree mode and writing the answer in radians

Where it slips in: A student computes arctan(2) with the calculator set to DEG, gets 63.435, and writes "arctan(2)=63.435 rad."

Don't do this: Drop the unit label or assume the calculator's mode matches what the problem wants.

3. Assuming arctan(tan(θ))=θ for every θ

Where it slips in: A student uses a value outside of the principal range and cancels the functions incorrectly.

The correct way: arctan(tan(θ))=θ only when θ∈(−π/2,π/2).

A Brief Word on Who Computed This First

Madhava of Sangamagrama discovered the arctangent power series around 1400 CE. His school used it to compute π to 11 decimal places by evaluating 4arctan(1). The same series, applied to arctan(1/2) and combined with the reflection identity arctan(2)=π/2−arctan(1/2), gives the modern decimals of arctan(2) as fast as any 21st-century algorithm.

Frequently Asked Questions

Is arctan 2 a rational or irrational number?
Irrational. It is also transcendental.

What is arctan 2 in radians?
arctan(2)≈1.1071 rad.

What is arctan 2 in degrees?
arctan(2)≈63.435°.

Is arctan 2 the same as cot 2?
No, they share no relationship beyond the coincidence of using the digit 2.

Can arctan 2 be written using pi?
Not as a rational multiple of π.

Why does arctan have a principal range of (−π/2,π/2)?
The tangent function is many-to-one — restricting arctan's output to that range picks the principal solution.