Arctan 0 — Value in Degrees and Radians
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Arctan 0 — Value in Degrees and Radians
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.
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Bhanzu Team 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
- Find arctan((\frac{1}{\sqrt{3}})) in degrees and radians.
- Evaluate arctan 0 + arctan(-1).
- 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.