Arctangent Calculator
Enter any number to find its arctangent, the angle θ = tan⁻¹(x), in degrees or radians.
Updated
Arctangent Calculator
Our Arctangent Calculator finds the angle whose tangent is a given number, arctan(x) or tan⁻¹(x). Type any number, a fraction such as 4/3 or a root such as √3 and the angle appears straight away in degrees or radians, to 10 decimal places. Common values give exact answers (arctan(1) = 45° = π/4), and the calculator also checks the result and lists the other angles with the same tangent. Unlike arcsine and arccosine, arctangent works for every real number.
Quick answer
The arctangent of x is the angle whose tangent is x, between −90° and 90°. For example, arctan(1) = 45° because tan 45° = 1. Enter any number above and the calculator shows the angle instantly.
- Formula: if tan θ = x, then θ = arctan(x) = tan⁻¹(x)
- Key values: arctan(0) = 0°, arctan(√3/3) = 30°, arctan(1) = 45°, arctan(√3) = 60°
- Domain: every real number
- Range: strictly between −90° and 90° (−π/2 and π/2), never reaching either
What Is Arctangent?
Arctangent is the inverse of the tangent function. Tangent takes an angle and gives a ratio; arctangent takes the ratio back to the angle. In a right triangle, if you know the side opposite an angle and the side next to it, arctangent gives the angle:
θ = arctan(opposite ÷ adj)
It is written arctan x, tan⁻¹ x or atan x. The ⁻¹ means "inverse function", not a power: tan⁻¹ x is not 1 ÷ tan x (that is the cotangent).
Arctangent Table of Common Values
| x | Degrees | Radians |
|---|---|---|
| −√3 ≈ −1.7321 | −60° | −π/3 |
| −1 | −45° | −π/4 |
| −√3/3 ≈ −0.5774 | −30° | −π/6 |
| 0 | 0° | 0 |
| 2 − √3 ≈ 0.2679 | 15° | π/12 |
| √3/3 ≈ 0.5774 | 30° | π/6 |
| 1 | 45° | π/4 |
| √3 ≈ 1.7321 | 60° | π/3 |
| 2 + √3 ≈ 3.7321 | 75° | 5π/12 |
√3/3 is the same number as 1/√3. Other values have no simple exact angle, for example arctan(0.5) ≈ 26.5651° and arctan(2) ≈ 63.4349°. As x grows, arctan(x) creeps toward 90°: arctan(100) ≈ 89.4271°.
The Arctangent Graph
- Shape: a rising S-curve through (0, 0°), flattening out on both sides.
- Always increasing: a bigger x always gives a bigger angle.
- Horizontal asymptotes: y = 90° and y = −90°, approached but never reached.
- Odd function: arctan(−x) = −arctan x, symmetric about the origin.
- Steepest at 0: near x = 0 the angle in radians is almost equal to x.
Why Arctangent Gives One Angle
Tangent repeats every 180°, so many angles share the same tangent: tan 45° = tan 225° = 1. A function must give one answer, so arctangent returns the principal value, the one between −90° and 90°. Every angle with tan θ = x is:
For tan θ = 1 between 0° and 360° that gives 45° and 225°. The calculator shows the second angle in its "Other angles" row.
Arctan, Slopes and Atan2
The angle of any slope is the arctangent of rise over run. A line with slope m makes an angle of arctan(m) with the horizontal.
Arctan alone cannot tell (3, 4) from (−3, −4), because 4/3 and −4/−3 are the same ratio. Programs therefore have atan2(y, x), which looks at the signs of both parts and returns a direction anywhere from −180° to 180°:
atan2(4, −3) ≈ 126.87°
Use atan2 for the direction of a vector or a point; use arctan for a ratio such as a slope or a triangle angle.
Arctangent Rules and Identities
- Inverse: tan(arctan x) = x for every real x.
- Only one way round: arctan(tan θ) = θ only for −90° < θ < 90°; arctan(tan 225°) = 45°.
- Negative values: arctan(−x) = −arctan x.
- Reciprocal: arctan x + arctan(1/x) = 90° for x > 0 (−90° for x < 0).
- With arccot: arctan x + arccot x = 90° (π/2).
- Calculus: the derivative of arctan x is 1 ÷ (1 + x2), and π = 4 × arctan(1).
Worked Examples
Roof pitch. A roof rises 6 inches for every 12 inches across (a 6/12 pitch):
= arctan(0.5)
≈ 26.57°
Road gradient. An 8% grade rises 8 m per 100 m across, so the angle is arctan(0.08) ≈ 4.57°, not 8°.
Steep and negative values. arctan(10/8) ≈ 51.3402°, arctan(−12) ≈ −85.2364° and arctan(−4/6) ≈ −33.6901°. A negative tangent always gives a negative angle.
Arctangent in Excel and Code
| Tool | arctan(1) in degrees |
|---|---|
| Excel / Google Sheets | =DEGREES(ATAN(1)) |
| Python | math.degrees(math.atan(1)) |
| JavaScript | Math.atan(1) * 180 / Math.PI |
| Scientific calculator | DEG mode, then SHIFT tan (tan⁻¹) 1 |
ATAN, atan and Math.atan return radians, so convert to degrees as shown. Watch the argument order of atan2: Excel uses =ATAN2(x, y), while Python and JavaScript use atan2(y, x).
Common Mistakes
- Reading tan⁻¹ as 1 ÷ tan. tan⁻¹ x is an angle; 1 ÷ tan θ is the cotangent.
- Losing the quadrant. arctan(4/−3) is −53.13°; the direction of (−3, 4) is 126.87° (use atan2).
- Treating a percent grade as degrees. A 100% grade is 45°, not 100°.
- Expecting exactly 90°. No finite number has an arctangent of 90°.
- Wrong angle mode. arctan(1) is 45 in degrees but 0.7854 in radians.
Key Terms
- Inverse
- A function that undoes another; arctangent undoes tangent.
- Range
- The possible outputs; for arctangent, between −90° and 90°.
- Asymptote
- A line the graph approaches but never reaches; y = ±90° here.
- Atan2
- A two-argument arctangent that keeps the quadrant.
- Arccot
- The inverse of cotangent; arctan x + arccot x = 90°.
- Cotangent
- 1 ÷ tan θ, often confused with tan⁻¹.
How to Use the Arctangent Calculator
- Enter the value. Type the tangent value x, which can be any number, or press one of the buttons (√3/3, 1, √3 and −1). You can type a decimal such as 0.5, a fraction such as 4/3 or 6/12, or a root such as √3 or 1/√3.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad) for the answer. The table below the result always shows both.
- Read and copy the result. The angle θ = arctan(x) appears as you type, to 10 decimal places, with the exact angle for common values (for example arctan(√3) = 60° = π/3 rad). The table also checks the answer and gives the other angle with the same tangent. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter −1 and the result is −45°, or −π/4 rad. The table shows that tan(−45°) = −1 and that 135° has the same tangent.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is arctangent?
Arctangent is the inverse of tangent. arctan(x), also written tan−1(x), is the angle between −90° and 90° whose tangent is x. For example arctan(1) = 45°, because tan 45° = 1.
What is arctan 0, arctan 1, arctan √3 and arctan(−1)?
arctan(0) = 0°, arctan(√3/3) = 30° (π/6), arctan(1) = 45° (π/4), arctan(√3) = 60° (π/3) and arctan(−1) = −45° (−π/4).
What is arctan of infinity?
There is no number called infinity to put in, but as x grows without limit arctan(x) approaches 90° (π/2), and as x falls toward minus infinity it approaches −90°. It never reaches either: arctan(100) ≈ 89.4271°.
What are the domain and range of arctangent?
The domain is every real number, because tangent takes every value. The range is strictly between −90° and 90° (−π/2 and π/2); the end points are never reached.
Is tan⁻¹(x) the same as 1/tan(x)?
No. tan−1(x) is the inverse function and returns an angle: tan−1(1) = 45°. 1 ÷ tan(x) is the cotangent, cot x: cot 45° = 1, and cot 30° = √3 ≈ 1.7321.
How do I find all angles with a given tangent?
Tangent repeats every 180°, so use θ = arctan(x) + 180° × k for any whole number k. For tan θ = 1 between 0° and 360° that gives 45° and 225°.
What is arctan(0.5) in degrees?
arctan(0.5) ≈ 26.5651°, or about 0.4636 radians. It has no simple exact form. It is the angle of a 6/12 roof pitch, and 206.5651° has the same tangent.
How do you calculate arctangent in Excel or on a calculator?
In Excel use =DEGREES(ATAN(1)) for degrees or =ATAN(1) for radians. On a scientific calculator set DEG mode and press SHIFT then tan (the tan−1 key). In code use Math.atan(x) in JavaScript or math.atan(x) in Python, both in radians.
What is the difference between atan and atan2?
atan(x) takes one ratio and returns an angle from −90° to 90°. atan2(y, x) takes the two parts separately, keeps their signs and returns a direction from −180° to 180°. For the point (−3, 4), atan(4 ÷ −3) ≈ −53.13° but atan2(4, −3) ≈ 126.87°.
Is the Arctangent Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Arctangent Calculator to find the angle for any tangent value in degrees or radians instantly.