Inverse Tangent Calculator (Arctan)
The inverse tangent of 1 is 45 degrees, because a 45 degree angle has a tangent of exactly 1. This inverse tangent calculator (also written arctan or tan-1) takes any ratio and returns the angle behind it, in both degrees and radians. Feed it a slope, an opposite-over-adjacent ratio, or any real number, positive or negative.
Quick answer
Tangent turns an angle into a ratio. Inverse tangent goes the other way, from ratio back to angle.
Angle
45 degrees
Radians
0.785398
Radians / pi
0.25
What this tells you
- •Tangent turns an angle into a ratio. Inverse tangent goes the other way, from ratio back to angle.
- •Arctan accepts any real number and always answers with an angle between -90 and 90 degrees.
- •Arctan of 0 is 0 degrees, arctan of 1 is 45 degrees, and very large inputs approach 90 degrees.
- •Negative inputs give negative angles. Arctan of -1 is -45 degrees.
How to Use
- 1Enter the ratio or number you have, like opposite divided by adjacent from a right triangle.
- 2Read the angle in degrees as the main result.
- 3The radians value and its multiple of pi are shown below for math homework that wants exact form.
- 4For a slope, enter rise divided by run to get the slope angle.
How It Works
Formula
angle = arctan(x), where tan(angle) = xArctan answers the question, which angle has this tangent? Since tan(45 degrees) = 1, arctan(1) = 45 degrees. The function only returns principal values between -90 and 90 degrees, because tangent repeats every 180 degrees and a single answer needs a chosen range. A ramp that rises 1 foot over a 2 foot run has a slope ratio of 0.5, and arctan(0.5) says the ramp angle is about 26.57 degrees.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Arctan of 1
The diagonal of a square makes a 45 degree angle, and its tangent is exactly 1.
Slope angle of a ramp
A rise of 1 over a run of 2 gives a ratio of 0.5, which is an angle of about 26.57 degrees.
Arctan of a negative number
Negative ratios return negative angles, here about -68.2 degrees or -1.19 radians.
Common Arctan Values
Inputs with clean angle answers, in degrees and radians.
| x | arctan(x) degrees | arctan(x) radians |
|---|---|---|
| 0 | 0 | 0 |
| 0.5774 | 30 | pi/6 |
| 1 | 45 | pi/4 |
| 1.7321 | 60 | pi/3 |
| -1 | -45 | -pi/4 |
| 1000 | 89.94 | close to pi/2 |
Common mistakes
- Reading the result in the wrong unit. 45 degrees and 0.785 radians are the same angle, and calculators in the wrong mode cause most arctan errors.
- Expecting angles outside -90 to 90 degrees. Arctan only returns principal values, so an angle of 135 degrees comes back as -45.
- Confusing arctan with 1/tan. The -1 in tan-1 means inverse function, not reciprocal. The reciprocal of tangent is cotangent.
- Using arctan alone to find direction between two points. That needs the two-argument atan2 form, which keeps track of the quadrant.