Arctan 2 — Value, Radians, Degrees, Worked Examples

The arctangent of $2$ is approximately $1.1071$ radians or $63.435°$.

Quick Answer:

Result: $\arctan(2) \approx 1.1071487$ radians $\approx 63.434965°$

Notation: $\arctan(2) = \tan^{-1}(2)$ — read as "the angle in $(-\pi/2, \pi/2)$ 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 $\pi$ and is not constructible from radicals. The value is irrational and transcendental.

Quick Reference — Arctan at Nearby Integer Inputs

The arctan function approaches $\pi/2$ ($90°$) as $x \to +\infty$ — it never reaches it. So $\arctan(2) \approx 63.4°$ 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 $2$ needs to be converted into an inclination angle. The angle a road grade of $200%$ makes with the horizontal is $\arctan(2) \approx 63.4°$. The angle the long diagonal of a $1 \times 2$ rectangle makes with the long side is also $\arctan(2)$ — by the right-triangle definition, the rise is $1$, the run is $2$, but the angle off the short side has tangent $2/1 = 2$. The vector $(1, 2)$ in the plane makes an angle of $\arctan(2)$ with the positive $x$-axis. 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 $\theta$ in the open interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ such that $\tan\theta = x$. For $x = 2$:

$$\arctan(2) = \theta \iff \tan\theta = 2 ;\text{ with }; \theta \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right).$$

Because tangent is strictly increasing on this interval and runs from $-\infty$ to $+\infty$, exactly one $\theta$ satisfies the equation. That $\theta$ is $\arctan(2) \approx 1.1071$ rad.

In degree-mode, the interval is $(-90°, 90°)$ and the answer is $\approx 63.435°$.

Why Arctan 2 Has No Clean Exact Form

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

A theorem due to Niven says: if $\theta$ is a rational multiple of $\pi$ and $\tan\theta$ is rational, then $\tan\theta \in {0, \pm 1, \text{undef}}$. So $\tan\theta = 2$ forces $\theta$ to be an irrational multiple of $\pi$. There is no expression for $\arctan(2)$ as $p\pi/q$ for any integers $p$ and $q$.

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 (the workhorse)

Switch the calculator to radian mode, press the $\tan^{-1}$ or $\arctan$ key, and enter $2$. The display reads $1.1071487\ldots$ rad.

Switch to degree mode and the same calculation reads $63.434965\ldots°$.

Conversion between the two:

$$\arctan(2){\deg} = \arctan(2){\rm rad} \times \dfrac{180}{\pi} = 1.1071487 \times \dfrac{180}{\pi} \approx 63.43°.$$

Final answer: $\arctan(2) \approx 1.1071$ rad $\approx 63.435°$.

Method 2: Madhava–Gregory series (numerical, by hand for understanding)

The Madhava–Gregory series for arctangent is:

$$\arctan(x) = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \cdots$$

This converges only for $|x| \leq 1$. Substituting $x = 2$ directly gives a divergent series — do not do this. Use the reflection identity instead:

$$\arctan(2) = \dfrac{\pi}{2} - \arctan!\left(\dfrac{1}{2}\right).$$

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

$$\arctan(1/2) = \dfrac{1}{2} - \dfrac{1}{24} + \dfrac{1}{160} - \dfrac{1}{896} + \cdots \approx 0.4636.$$

So $\arctan(2) \approx \dfrac{\pi}{2} - 0.4636 = 1.5708 - 0.4636 = 1.1072$ rad.

In degrees: $90° - 26.565° = 63.435°$.

Final answer: $\arctan(2) \approx 1.1072$ rad $\approx 63.435°$ (matches Method 1 to four decimal places).

Method 3: Right-triangle reading (geometric)

Build a right triangle with legs of length $1$ and $2$. The hypotenuse is $\sqrt{1^2 + 2^2} = \sqrt{5}$. The angle opposite the leg of length $2$ has:

$$\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{2}{1} = 2.$$

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

$$\sin(\arctan(2)) = \dfrac{2}{\sqrt{5}} = \dfrac{2\sqrt{5}}{5} \approx 0.8944,$$

$$\cos(\arctan(2)) = \dfrac{1}{\sqrt{5}} = \dfrac{\sqrt{5}}{5} \approx 0.4472.$$

The same numbers fall out of the unit-circle reading — the point on the unit circle at angle $\arctan(2)$ is $\left(1/\sqrt{5}, 2/\sqrt{5}\right) \approx (0.4472, 0.8944)$.

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

Common Mistakes With Arctan 2

1. Reading "$\tan^{-1} 2$" as "$1/\tan 2$"

Where it slips in: A student sees $\tan^{-1}(2)$ on a calculator and types $1 \div \tan(2)$ 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: $\tan^{-1}(2) = \arctan(2) \approx 1.1071$ rad. The reciprocal of tangent is the cotangent, written $\cot(\theta)$ or $1/\tan(\theta)$. These two values differ wildly: $\cot(2 \text{ rad}) \approx -0.457$ versus $\arctan(2) \approx 1.107$ 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.

The correct way: $63.435°$ and $1.1071$ rad are the same angle in different units. Always check the calculator's mode before reading the screen.

3. Assuming $\arctan(\tan(\theta)) = \theta$ for every $\theta$

Where it slips in: A student is told $\tan(2.5 \text{ rad}) \approx -0.7470$, then asked for $\arctan(-0.7470)$, and writes $2.5$ rad.

Don't do this: Cancel the functions without checking that the input lies in the principal range $(-\pi/2, \pi/2)$.

The correct way: $\arctan(\tan(\theta)) = \theta$ only when $\theta \in (-\pi/2, \pi/2)$. For $\theta = 2.5$ rad (which is outside this range), $\arctan(\tan(2.5)) = 2.5 - \pi \approx -0.6416$ rad. The arctan output is always in the principal range — never outside it.

A Brief Word on Who Computed This First

Madhava of Sangamagrama (c. 1340 – c. 1425, India) discovered the arctangent power series — the one that today bears his name jointly with James Gregory — around 1400 CE. Madhava's school in Kerala used it to compute $\pi$ to 11 decimal places by evaluating $4 \arctan(1)$. The same series, applied to $\arctan(1/2)$ and combined with the reflection identity $\arctan(2) = \pi/2 - \arctan(1/2)$, gives the modern decimals of $\arctan(2)$ as fast as any 21st-century algorithm.