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# Arctan 0 — Value in Degrees and Radians

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

TL;DR  
Arctan 0 equals 0° or 0 radians — the angle in \((-\frac{\pi}{2}, \frac{\pi}{2})\) whose tangent is 0. This article gives the value in both units, the unit-circle reason it is 0 and not \(\pi\), a tan-inverse reference table, two worked methods, the mistakes around the restricted range, and FAQs.

BT  
[Bhanzu Team](/content/authors/bhanzu-team/index.html) Last updated on June 13, 2022 6 min read

## Arctan 0 in Degrees and Radians

The value of **arctan 0** is 0°, which is also 0 radians. It is the unique angle inside the inverse-tangent range \((-\frac{\pi}{2}, \frac{\pi}{2})\) whose tangent equals 0.

### Quick Answer:

- Result: arctan 0 = 0° = 0 radians
- Notation: arctan 0 = tan⁻¹ 0 (the inverse function, NOT 1/tan)
- Method shown: read the angle whose tangent is 0 from the unit circle / special-angle table
- Exact form: 0 (the same value in both degrees and radians)
- Approximate value: 0 (exact, no rounding)

Tangent is \(\frac{\sin}{\cos}\), so it is zero exactly when sine is zero and cosine is not. The angle in the inverse-tangent range where that happens is 0 itself — the positive x-axis direction on the unit circle.

## Tan-Inverse Reference Table

These are the tan-inverse values readers look up around arctan 0. Each is shown in radians and degrees, with the tangent value that produces it.

| Input | arctan x (rad) | arctan x (deg) | Note |
| --- | --- | --- | --- |
| -1 | -\frac{\pi}{4} | -45° | tangent -1 |
| -\frac{1}{\sqrt{3}} | -\frac{\pi}{6} | -30° | tangent -1/3 |
| 0 | 0 | 0° | tangent 0 |
| \frac{1}{\sqrt{3}} | \frac{\pi}{6} | 30° | tangent 1/3 |
| 1 | \frac{\pi}{4} | 45° | tangent 1 |
| \sqrt{3} | \frac{\pi}{3} | 60° | tangent \sqrt{3} |
| → +\infty | → \frac{\pi}{2} | → 90° | horizontal asymptote (never reached) |

## Where Arctan 0 Appears

The angle 0 marks a flat, level direction — zero slope. **Is arctan 0 equal to 0 or pi?** It is 0, and that distinction is what makes inverse tangent the standard tool for recovering a heading from a slope. In computer graphics and navigation, `atan2(y, x)` returns the angle of a vector; when y=0 and x>0, that angle is arctan 0=0 — pointing due east. In calculus, arctan x is the antiderivative of \(\frac{1}{1 + x^2}\), and evaluating it at 0 gives the lower limit of the integral that defines \(\frac{\pi}{4}\). A road with zero rise over run has an incline of arctan 0=0° — perfectly level.

## What Arctan Means

Arctangent answers "which angle has this tangent?" Written arctan x or tan⁻¹ x, it is the inverse of the tangent function. Tangent repeats every \(\pi\) and is many-to-one — tan 0=0, but tan \(\pi\)=0 too — so to define a single-valued inverse, mathematicians restrict tangent to the open interval \((-\frac{\pi}{2}, \frac{\pi}{2})\), where it rises steadily from -∞ to +∞. The inverse of that restricted tangent is the arctangent, with outputs confined to that interval.  
Feed it 0 and it returns the centre of that range: 0.

## How to Find Arctan 0

### **Method 1: Read it from the special-angle table**

The tangent table gives tan 0°=0. Arctangent reverses the lookup:

arctan 0=0°=0 rad.  
Although tan \(\pi\)=0 as well, only 0 lies inside the arctangent range, so the answer is unique.

**Final answer:** arctan 0=0°=0 rad.

### **Method 2: Use the ratio on the unit circle**  
Tangent is the ratio \(\frac{\sin\theta}{\cos\theta} = \frac{y}{x}\). Setting this to 0 needs y=0 with x≠0. Inside \((-\frac{\pi}{2}, \frac{\pi}{2})\), the only such point is (1,0), at angle 0:

arctan 0=\(\frac{0}{1} \to \theta = 0.\)  
**Final answer:** the same 0°=0 rad, confirmed by the unit-circle ratio.

## Examples of Arctan 0

### Example 1
**State arctan 0 in both degrees and radians.**
From the special-angle table, tan 0=0, so arctan 0=0°=0 rad. The value is identical in both units.
**Final answer:** 0°=0 rad.

### Example 2
**A student writes arctan 0=\(\pi\) because tan \(\pi\)=0. Check it.**
_Wrong attempt._ Reasoning that any angle with tangent 0 qualifies, the student notes tan \(\pi\)=0 and reports arctan 0=\(\pi\) (that is, 180°).
_The break._ Arctangent returns only angles in \((-\frac{\pi}{2}, \frac{\pi}{2})\). The value \(\pi\) sits well outside that interval, so it cannot be an arctangent output — even though its tangent is indeed 0. Returning \(\pi\) would break the function's own range, the way two angles can share a tangent but only one is the principal value.
_Correct._ The angle with tangent 0 that lies inside the range is 0:

arctan 0=0.
**Final answer:** arctan 0=0, not \(\pi\).

### Example 3
**Evaluate arctan 0+arctan 1.**
arctan 0=0 and arctan 1=\(\frac{\pi}{4}\), so the sum is \(\frac{\pi}{4} = 45°\).
**Final answer:** \(\frac{\pi}{4} = 45°\).

### Example 4
**Find the limit \(\lim_{x \to 0} \arctan x\).**
Arctangent is continuous everywhere, so the limit equals the function value at 0: arctan 0=0.
**Final answer:** 0.

### Example 5
**A vector points along (5,0). What angle does it make with the positive x-axis?**
The angle is arctan(0/5)=arctan 0=0°. The vector lies flat along the axis — zero elevation.
**Final answer:** 0°=0 rad.

## Where Tan-Inverse Trips Students Up

### Mistake 1: Returning \(\pi\) instead of 0
**Where it slips in:** A reader notices tan \(\pi\)=0 and reports arctan 0=\(\pi\).
**Don't do this:** Pick any angle whose tangent is 0 without checking the range.
**The correct way:** Arctangent outputs only angles in \((-\frac{\pi}{2}, \frac{\pi}{2})\), so the principal value is 0.

### Mistake 2: Reading tan⁻¹ 0 as a reciprocal
**Where it slips in:** A reader treats tan⁻¹ 0 as 1/tan 0.
**Don't do this:** Read the -1 as an exponent — which here would force a division by 0.
**The correct way:** tan⁻¹ is the inverse function, arctan. The reciprocal of tangent is cot x=1/tan x, governed by the reciprocal identities — and cot 0 is undefined, a different statement entirely from arctan 0=0.

### Mistake 3: Confusing arctan 0 with the asymptote
**Where it slips in:** A reader mixes up the input 0 (which gives angle 0) with the large-input behaviour (which approaches 90°).
**Don't do this:** Assume arctangent of a small input is near its maximum.
**The correct way:** arctan x → \(\frac{\pi}{2}\) only as x → +∞; at x=0 the output is exactly 0, the centre of the range. The graph passes through the origin.

## What to Remember About Arctan 0

- **Arctan 0** equals 0° or 0 radians — the angle whose tangent is exactly 0.
- On the unit circle it is the direction (1,0) along the positive x-axis, where \(\frac{y}{x} = 0\).
- Although tan \(\pi\)=0, arctan 0 is 0 because the range is \((-\frac{\pi}{2}, \frac{\pi}{2})\).
- tan⁻¹ is the inverse function, not the reciprocal 1/tan.

## Practice These Three

1. Find arctan(\(\frac{1}{\sqrt{3}}\)) in degrees and radians.
2. Evaluate arctan 0 + arctan(-1).
3. A vector points along (0,4). Why does arctan(4/0) fail, and what angle is the vector actually at?

If Problem 2 gives -\(\frac{\pi}{4}\), the arctan 0 term contributes nothing, leaving arctan(-1).

## Frequently Asked Questions

**What is arctan 0?**  
The angle whose tangent is 0, namely 0° or 0 radians.

**Is arctan 0 equal to 0 or pi?**  
0. Although tan \(\pi\)=0, only 0 lies in the arctangent range \((-\frac{\pi}{2}, \frac{\pi}{2})\).

**What is arctan 0 in radians?**  
0 radians — the same value as 0°.

**Is tan inverse 0 the same as 1/tan 0?**  
No. arctan 0=0 is the inverse function; 1/tan 0=cot 0 is the reciprocal, and cot 0 is undefined.

**Why is arctan 0 not 180 degrees?**  
Because 180°=\(\pi\) is outside the principal range of arctangent. The function returns the unique angle inside \((-\frac{\pi}{2}, \frac{\pi}{2})\), which is 0.