Inverse Tangent Calculator (Arctan)
The inverse tangent of 1 is 45 degrees, or 0.785398 radians. This inverse tangent calculator, also called an arctan or tan inverse calculator, converts any finite real tangent ratio into its principal angle. It reports degrees, radians, and radians divided by pi. In a right triangle, the tangent ratio is opposite side divided by adjacent side. A rise of 1 over a horizontal run of 2 gives a ratio of 0.5, and arctan(0.5) gives an angle of about 26.5651 degrees. The same calculation can describe the inclination of a line when rise and run use the same unit. Arctan returns one principal answer between -90 and 90 degrees. Tangent repeats every 180 degrees, so other angles can share the same tangent. A single ratio also loses quadrant information when it comes from separate horizontal and vertical coordinates. Use an atan2 calculation when both coordinate signs must determine a full direction.
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.
- •The result approaches 90 degrees for very large positive inputs and -90 degrees for very large negative inputs, but never reaches either endpoint for a finite input.
- •The displayed degree angle is rounded to 4 decimal places, while the secondary degree and radian values are rounded to 6 decimal places.
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.
- 5Keep rise and run in the same unit before dividing so the input is a dimensionless ratio.
- 6Check whether your problem needs a principal inclination or a full-circle direction. Arctan alone cannot identify all four quadrants.
- 7Use the radians-over-pi result as a decimal multiple of pi, not automatically as an exact fraction.
How It Works
Formula
angle = arctan(x), where tan(angle) = xArctan answers which principal angle has tangent x. Since tan(45 degrees) = 1, arctan(1) = 45 degrees. The code first uses Math.atan(x), which returns radians in the interval greater than -pi/2 and less than pi/2. It converts radians to degrees with radians x 180 / pi. For a rise of 1 over a run of 2, x = 1 / 2 = 0.5. Math.atan(0.5) is about 0.463647609 radians, and multiplying by 180 / pi gives about 26.565051 degrees. The output rounds degrees and radians to 6 decimal places, radians divided by pi to 6 decimal places, and the formatted degree result to 4 decimal places.
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 the calculator displays as 26.5651 degrees.
Arctan of a negative number
Negative ratios return negative angles. Arctan(-2.5) is about -68.1986 degrees or -1.19029 radians.
Arctan of 0
A zero tangent ratio has no vertical change relative to horizontal change. Math.atan(0) returns 0 radians, which is also 0 degrees.
Angle for a 10% grade
A 10% grade means rise divided by horizontal run equals 0.1. Arctan(0.1) is about 0.099669 radians or 5.7106 degrees.
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 |
Why arctan needs a principal range
Tangent has a period of 180 degrees. The tangent of 45 degrees equals 1, but the tangent of 225 degrees also equals 1. An inverse function must return one output for each input, so arctan selects the principal angle strictly between -90 and 90 degrees.
That range works well for line inclination and right-triangle ratios with a positive adjacent length. It is incomplete for a directed line or coordinate vector. The ratios y/x for points in opposite quadrants can match because both signs cancel. Atan2 uses x and y separately and can return the correct quadrant.
Radians divided by pi can reveal familiar angles. For x = 1, the result is 0.25 because pi/4 divided by pi equals one quarter. Most inputs do not produce a simple exact fraction, so a displayed decimal such as 0.147584 should be treated as rounded.
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.
- Entering an angle when the tool expects a tangent ratio. To evaluate tangent from an angle, use a trigonometry calculator instead.
- Using run divided by rise for a slope angle measured from horizontal. The usual input is rise divided by horizontal run.
- Treating a 10% grade as 10. A 10% grade is the decimal ratio 0.1 and gives about 5.7106 degrees.
- Assuming the radians-over-pi decimal is an exact fraction. It is rounded to 6 decimal places unless the value happens to map cleanly.
Embed this calculator on your site
Drop this single line where you want the calculator to appear. It is responsive, mobile-friendly, resizes automatically, and is free to use with attribution.
<script src="https://calctide.com/embed.js" data-tool="inverse-tangent-calculator" async></script>Preview the embed at /embed/inverse-tangent-calculator/.