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Arctan Calculator

Arctan calculator finds the angle from opposite and adjacent sides using inverse tangent. Opposite 3 and adjacent 4 gives theta = 36.87 degrees. Used for slope angles, ramp grades, and right triangle problems.

Enter Values

Opposite Side

Adjacent Side

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

\theta = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)

How it works: Arctan (inverse tangent) returns the angle whose tangent equals opposite divided by adjacent. Result is between -90 and 90 degrees.

Worked Example

Find the angle of a right triangle with opposite side 3 and adjacent side 4.

1Step 1: ratio = opposite / adjacent = 3/4 = 0.75.

2Step 2: theta = arctan(0.75) = 36.87 degrees = 0.6435 radians.

3Step 3: Verify: tan(36.87) = 0.75.

4Step 4: The hypotenuse = sqrt(3^2 + 4^2) = sqrt(25) = 5 (a 3-4-5 right triangle). This is a classic Pythagorean triple.

5Step 5: Practical use: a ramp rising 3 ft over 4 ft horizontal = 36.87 degree incline = 75% grade.

How Inverse Tangent (Arctan) Works

Arctan is the inverse of the tangent function. Given the ratio of opposite to adjacent sides in a right triangle, it returns the angle.

theta = arctan(opposite/adjacent). For a 3-4-5 triangle: arctan(3/4) = 36.87 degrees
Output range is -90 to 90 degrees. Arctan always returns an angle in this range
Key values: arctan(0) = 0, arctan(1) = 45, arctan(sqrt(3)) = 60 degrees
Slope grade (%) = (rise/run) x 100 = tan(angle) x 100. A 45-degree slope is a 100% grade

Arctan is the most commonly used inverse trig function in engineering, surveying, and physics because most real-world angle problems involve rise-over-run ratios.

You can also calculate changes using our Tangent Calculator, Sine Calculator, Cosine Calculator or Arccos Calculator.

Frequently Asked Questions

What angle do opposite 3 and adjacent 4 give?

arctan(3/4) = arctan(0.75) = 36.87 degrees (0.6435 radians). This is one angle of the classic 3-4-5 right triangle. The other acute angle is 90 - 36.87 = 53.13 degrees.

What is arctan(1) in degrees?

arctan(1) = 45 degrees exactly. This means when the opposite and adjacent sides are equal, the angle is 45 degrees. It corresponds to a 100% slope grade.

How do I find the angle of a slope or ramp?

Measure the rise (vertical) and run (horizontal), then calculate arctan(rise/run). A ramp that rises 1 foot over 12 feet horizontal: arctan(1/12) = 4.76 degrees. ADA-compliant ramps require a maximum slope of 1:12 (4.76 degrees).

What angle is a 10% road grade?

A 10% grade means 10 ft rise per 100 ft run. arctan(10/100) = arctan(0.1) = 5.71 degrees. Other common grades: 6% = 3.43 degrees, 15% = 8.53 degrees, 45% = 24.23 degrees.

Can the adjacent side be zero?

No. When the adjacent side is zero, the ratio becomes infinite, and the angle is exactly 90 degrees (a vertical line). The arctan function approaches but never reaches 90 degrees. For truly vertical slopes, the angle is 90 degrees by definition.

Is it possible to embed the Arctan Calculator on another website?

Yes, embedding the Arctan Calculator is free. Hit the "Embed" button on this page, adjust the width, height, and theme, then grab the iframe code. It works on WordPress, Wix, Squarespace, Shopify, and plain HTML pages. No registration needed. Full instructions at calculory.com/services/embed-calculators.

AI Assistant

Ask about this calculator

I can help you understand the arctan calculator formula, interpret your results, and answer follow-up questions.

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