Arctan Calculator

Find the angle from the ratio of two sides of a right triangle.

45.00°

1.00Ratio (y/x)

0.785 radAngle (Radians)

45.00°Angle (Degrees)

What is Arctan?

Arctan, short for arctangent, is the inverse trigonometric function of the tangent. While the tangent function takes an angle and gives the ratio of the opposite side to the adjacent side in a right-angled triangle, arctan does the reverse: it takes the ratio and gives the angle. Arctan is also commonly written as tan⁻¹. The primary use of an arctan calculator is to find an unknown angle when you know the lengths of the two sides forming the right angle.

There’s a common misconception that you cannot use arctan in a browser calculator. This is incorrect. Modern web browsers can easily compute arctan using JavaScript’s built-in `Math.atan()` or `Math.atan2()` functions. This very page uses this functionality to provide instant calculations.

The Arctan Formula and Explanation

The formula to find an angle (θ) using arctan is straightforward. It is derived from the basic tangent definition.

θ = arctan(y / x)

Where ‘y’ is the length of the opposite side and ‘x’ is the length of the adjacent side. The result, θ, is the angle between the adjacent side and the hypotenuse. Our Right Triangle Calculator can help visualize this further.

Arctan Formula Variables

Variable	Meaning	Unit	Typical Range
θ	Calculated Angle	Degrees or Radians	-90° to +90° or -π/2 to +π/2 rad
y	Opposite Side Length	Any unit of length (m, ft, cm)	Any positive number
x	Adjacent Side Length	Any unit of length (m, ft, cm)	Any positive number

Practical Examples

Example 1: Calculating the Slope of a Ramp

Imagine you are building a wheelchair ramp that needs to rise 1 foot for every 12 feet of horizontal distance (a 1:12 slope, which is a common accessibility standard).

• Input (Opposite Side y): 1 ft
• Input (Adjacent Side x): 12 ft
• Calculation: θ = arctan(1 / 12) = arctan(0.0833)
• Result: The angle of the ramp is approximately 4.76 degrees.

Example 2: Angle of Elevation

You are standing 50 meters away from a tall tree. You measure the angle to the top of the tree from the ground. If the tree is 30 meters tall, what is the angle of elevation from where you stand?

• Input (Opposite Side y): 30 m
• Input (Adjacent Side x): 50 m
• Calculation: θ = arctan(30 / 50) = arctan(0.6)
• Result: The angle of elevation is approximately 30.96 degrees.

How to Use This Arctan Calculator

Using this calculator is simple and intuitive. Follow these steps for an accurate calculation:

1. Enter Opposite Side (y): Input the length of the side opposite to the angle you are trying to find.
2. Enter Adjacent Side (x): Input the length of the side that is adjacent to the angle.
3. Select Angle Unit: Choose whether you want the result in ‘Degrees’ or ‘Radians’ from the dropdown menu. The calculator updates automatically.
4. Interpret the Results: The main result is displayed prominently. You can also see intermediate values like the ratio and the angle in both units. For more complex trigonometric problems, you might want to use a Sine and Cosine Calculator.

Key Factors That Affect Arctan Calculations

Several factors influence the outcome and interpretation of an arctan calculation.

• The Ratio (y/x): This is the most critical factor. The value of the angle is entirely dependent on this ratio.
• Unit Selection (Degrees vs. Radians): The numerical value of the angle will be drastically different depending on the unit chosen. 180 degrees equals π (about 3.14159) radians.
• The Signs of x and y: In this basic calculator, we assume positive values for quadrant I. However, in full four-quadrant trigonometry, the signs determine if the angle is in the upper/lower or left/right half of the plane.
• Division by Zero: If the adjacent side (x) is 0, the ratio is undefined (infinite). The arctan of infinity is 90° or π/2 radians. Our calculator handles this case.
• Floating-Point Precision: Like all computer calculations, there can be tiny precision errors in the decimal places. This is usually negligible for practical applications.
• `atan` vs `atan2`: In programming, a function `atan2(y, x)` is often preferred over `atan(y/x)` because it correctly handles all four quadrants and the division-by-zero case automatically. Our tool uses this principle for robust calculations. A more advanced tool might expose this choice.

Frequently Asked Questions (FAQ)

What is arctan?

Arctan is the inverse function of tangent. It finds the angle corresponding to a given tangent ratio.

Is arctan the same as tan⁻¹?

Yes, arctan and tan⁻¹ represent the same inverse tangent function. It is important not to confuse tan⁻¹(x) with 1/tan(x), which is the cotangent function.

Why did I think I couldn’t use arctan in a browser?

This is a common misunderstanding. While simple calculator apps may not have an ‘arctan’ button, the underlying technology of all modern browsers (JavaScript) has full support for it, making tools like this one possible.

What’s the difference between degrees and radians?

They are two different units for measuring angles. A full circle is 360 degrees or 2π radians. You can learn more with our angle conversion tool.

What is the result of arctan(1)?

Arctan(1) is 45 degrees or π/4 radians. This occurs when the opposite and adjacent sides are equal.

What happens if the ‘Adjacent’ side is 0?

The angle approaches 90 degrees (or π/2 radians). The tangent is technically undefined, but the limit is well-defined.

Why does my scientific calculator give a different answer?

Your calculator is most likely set to a different angle mode (DEG, RAD, or GRAD). Ensure it matches the unit you are comparing against. To learn more about functions see our function graphing tool.

What is atan2?

In programming, `atan2(y, x)` is a two-argument function that calculates the arctangent and correctly determines the angle’s quadrant based on the signs of y and x, making it more robust than `atan(y/x)`. You can explore this in our quadrant calculator.

Related Tools and Internal Resources

• Right Triangle Calculator: Solves for all missing sides and angles of a right triangle.
• Sine and Cosine Calculator: Calculates the other primary trigonometric functions.
• Angle Conversion Calculator: Converts between degrees, radians, and other units.
• Quadrant Calculator: Helps determine the quadrant of an angle based on coordinates.