Arctan Calculator - Inverse Tangent (arctan)
Enter any real number to find its inverse tangent (arctan). Results appear in both degrees and radians with a full step-by-step breakdown. Switch to rise/run mode to compute the angle of a slope directly from two leg lengths. The interactive curve shows exactly where your result falls on the arctangent function.
Formula
Worked example
A ramp rises 3 m over a horizontal run of 4 m. The slope angle = arctan(3/4) = arctan(0.75) = 36.87 degrees (0.6435 radians). Verify: tan(36.87 deg) = 0.7500, confirming the result.
What is the arctan (inverse tangent) function?
Arctan, written arctan(x) or tan-1(x), is the inverse of the tangent function. While tan(angle) tells you the ratio of the opposite side to the adjacent side in a right triangle, arctan(ratio) tells you the angle that produces that ratio. The domain of arctan is all real numbers - you can take the arctan of any value from negative infinity to positive infinity. The output (range) is always between -90 and 90 degrees, or equivalently between -pi/2 and pi/2 radians. This restricted range is what makes arctan a proper function: for every input there is exactly one output.
How to calculate arctan
On a scientific calculator, press Shift (or 2nd) and then the Tan button to activate the tan-1 mode, enter your number, and press equals. On most computers and in most programming languages, the function is called atan(x) (or Math.atan(x) in JavaScript). To convert the radian result to degrees, multiply by 180/pi. To convert degrees back to radians, multiply by pi/180. The rise/run mode on this calculator is especially useful for engineers and builders: enter the height gained (rise) and horizontal distance (run) of a slope, and it computes the angle by forming the ratio and then applying arctan. For example, a 3-metre rise over 4 metres gives arctan(3/4) = 36.87 degrees.
Arctan identities and key properties
Several identities connect arctan to other inverse trig functions. The complementary identity arctan(x) + arccot(x) = pi/2 holds for all x. For negative inputs, arctan(-x) = -arctan(x), meaning the function is odd and symmetric about the origin. The derivative of arctan(x) with respect to x is 1/(1 + x^2), and its integral is x arctan(x) - (1/2)ln(1 + x^2) + C. The Maclaurin series arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... converges for |x| <= 1 and is the basis for several historical pi approximations, including Leibniz's series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...
Arctan vs. atan vs. atan2
All three names refer to the inverse tangent, but with different scopes. arctan(x) and atan(x) are two notations for the same single-argument function returning an angle in (-pi/2, pi/2). atan2(y, x) is a two-argument variant found in most programming languages that uses both the numerator (y = rise) and denominator (x = run) separately, rather than their ratio, to determine the correct quadrant. atan2 can return angles across the full range (-pi, pi], making it the right choice when you need to distinguish, for example, between a vector pointing right-and-up versus left-and-down. This calculator's rise/run mode is equivalent to atan2(rise, run) for run > 0.
Common arctan values
| x (tangent value) | arctan(x) in degrees | arctan(x) in radians | Fraction of pi |
|---|---|---|---|
| -infinity | -90 | -pi/2 | -pi/2 |
| -sqrt(3) ~ -1.7321 | -60 | -pi/3 | -pi/3 |
| -1 | -45 | -pi/4 | -pi/4 |
| -1/sqrt(3) ~ -0.5774 | -30 | -pi/6 | -pi/6 |
| 0 | 0 | 0 | 0 |
| 1/sqrt(3) ~ 0.5774 | 30 | pi/6 | pi/6 |
| 1 | 45 | pi/4 | pi/4 |
| sqrt(3) ~ 1.7321 | 60 | pi/3 | pi/3 |
| +infinity | 90 | pi/2 | pi/2 |
Exact and decimal arctan values for frequently encountered tangent inputs. Angles are in degrees and radians.
Frequently asked questions
What is arctan(1)?
arctan(1) = 45 degrees = pi/4 radians. This is because tan(45 deg) = 1, meaning a right triangle where the opposite and adjacent sides are equal has a base angle of 45 degrees.
What is the difference between arctan and tan-1?
They are the same function written in two different notations. arctan(x) uses prefix notation, while tan-1(x) uses superscript notation. Both mean "the angle whose tangent is x." The superscript -1 does NOT mean the reciprocal (1/tan); it means the inverse function. To avoid confusion, many mathematicians prefer arctan.
Why is the arctan output range limited to -90 to 90 degrees?
The tangent function is periodic, repeating every 180 degrees, so the same tangent value corresponds to infinitely many angles. To make arctan a proper function (one input, one output), mathematicians restrict the output to the interval (-90 deg, 90 deg), which is called the principal value. If you need an angle outside that range, you must add or subtract 180 degree increments depending on which quadrant your vector lies in, which is what the atan2 function does.
How do I find an angle of a slope using arctan?
Divide the rise (vertical change) by the run (horizontal change) to get the tangent of the slope angle, then apply arctan. For example, a road that rises 5 metres for every 20 metres of horizontal distance has a tangent of 5/20 = 0.25, and arctan(0.25) = 14.04 degrees. Use the rise/run mode on this calculator to enter the two distances directly without computing the ratio yourself.
What is arctan(0)?
arctan(0) = 0 degrees = 0 radians. This follows directly from tan(0) = 0.
Can I use arctan to find an angle in a right triangle?
Yes. In a right triangle, if you know the opposite and adjacent sides relative to the angle you want to find, the angle = arctan(opposite / adjacent). For example, if the opposite side is 7 and the adjacent is 10, the angle is arctan(7/10) = arctan(0.7) = 34.99 degrees. Enter the rise (opposite) and run (adjacent) directly in rise/run mode.