Arctan Calculator

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What is Arctan Calculator?

This arctan calculator turns a tangent ratio into an angle. If a ramp rises 1 metre over a horizontal distance of 4 metres, its tangent ratio is 1/4, and inverse tangent gives its inclination. The same operation appears when comparing vertical and horizontal components of a vector or solving a trigonometric equation.

The input is a number, not an angle. The page returns the principal angle in radians and degrees, rounded to four decimal places. Unlike inverse sine, inverse tangent accepts every finite real number. Its answer remains strictly between −90° and 90°, which is useful but also explains why a single ratio cannot identify every possible compass direction.

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How to Use This Calculator

Identify the opposite quantity and adjacent quantity, or the vertical change and horizontal run. Express both in the same unit before dividing.
Calculate their ratio and enter it in the Arctan field. A rise of 25 cm across a run of 100 cm becomes 0.25. If the slope is given as 25%, divide by 100 first.
Choose Calculate to see radians and degrees. Match the output unit to the formula, drawing or angle convention used in the rest of your work.
Check signs and the required quadrant before applying the result. Use Reset when starting a new example so an earlier ratio is not mistaken for the current one.

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Formula Used

θ = arctan(x). Degrees = arctan(x) × 180/π. For a slope, x = rise/run. The principal range is −π/2 < θ < π/2.

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Variables Explained

Item	Meaning
Arctan input x	A dimensionless tangent ratio, such as opposite side divided by adjacent side, or rise divided by horizontal run.
Radians	An angular unit with π radians in a straight angle. This output is appropriate for formulas that require radian measure.
Degrees	The same angle on a 360-degree full-turn scale. Positive inputs return positive principal angles and negative inputs return negative principal angles.
Run	The horizontal component of a slope example. It must be nonzero when calculating an ordinary rise/run ratio; a vertical line needs separate treatment.

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Worked Example

For a rise of 1 m and a run of 4 m, x = 1/4 = 0.25. The result is arctan(0.25) ≈ 0.2450 radians, or 14.0362°. This is also the angle corresponding to a 25% grade. The percentage describes a ratio, so a 25% grade does not mean a 25° inclination.

For x = 1, the returned angle is 0.7854 radians or 45°. For x = −1, it is −0.7854 radians or −45°. A much larger positive ratio, such as 10, gives about 84.2894°, approaching a right angle without reaching it for a finite input.

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Why Use This Calculator?

The sign tells you whether the principal angle lies above or below zero under your chosen convention. The magnitude describes how steep the ratio is. However, reversing both vector components leaves their quotient unchanged. A vector with components (1, 1) and another with components (−1, −1) both produce a ratio of 1, although they point in opposite directions.

When both coordinates and the full direction matter, use a two-argument atan2 function with the documented argument order. This page receives only one ratio and cannot recover the missing quadrant information. For a tangent equation, all solutions differ by integer multiples of 180°: θ = α + 180°k, where α is the principal answer. For positive right-triangle sides, the acute result usually resolves the geometry directly.

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Common Mistakes

Divide a percentage grade by 100 before entering it. Entering 25 instead of 0.25 describes a dramatically steeper slope.
Use horizontal run, rather than the sloping length, in rise/run. Rise divided by sloping length belongs to sine instead.
Do not divide by zero to describe a vertical line. Its angle must be handled from the geometry or vector components.
Do not round a small rise/run ratio to zero before using arctan; that can erase a shallow but meaningful inclination.

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Frequently Asked Questions

Why can I enter values outside −1 to 1?

Tangent can take any real value between its vertical asymptotes, so inverse tangent accepts any finite real ratio. The restricted −1 to 1 domain applies to real inverse sine and inverse cosine, not inverse tangent.

What is the difference between arctan and reciprocal tangent?

Arctan reverses the tangent function on its principal interval. Reciprocal tangent is cotangent, equal to 1/tan θ. The notation tan⁻¹ can mean inverse tangent in many textbooks, so read the surrounding context carefully.

How do I turn the answer back into a percentage slope?

Take the tangent of the angle using the proper degree or radian setting, then multiply the result by 100. For example, tan(45°) × 100 = 100%. A 100% grade is a 45° slope, rather than a vertical face.

Why does another program show 135°?

It may be resolving a vector in a different quadrant or giving a different branch of a tangent equation. Arctan(−1) returns −45° here; 135° has the same tangent but points in a different direction. Check the intended interval and signs.

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Sources & References

OpenStax — Precalculus 2e, inverse trigonometric functions