Arctan Calculator: How to Find Arctan
Arctan Calculator
Enter a numeric value to find its inverse tangent (arctan). You can get the result in degrees or radians.
This is a unitless ratio (e.g., opposite / adjacent).
Arctan Function Graph
What is Arctan?
Arctan, short for "arctangent," is the inverse function of the tangent. In trigonometry, while the tangent function takes an angle and gives you the ratio of the opposite side to the adjacent side in a right-angled triangle, arctan does the opposite. It takes that ratio as an input and gives you back the angle. It's a way to answer the question: "Which angle has this particular tangent?"
The function is commonly written as arctan(x) or tan-1(x). It is important not to confuse tan-1(x) with 1/tan(x), which is the cotangent function. Arctan is crucial in many fields like engineering, physics, navigation, and geometry for finding angles when side lengths are known.
The Arctan Formula and Explanation
The fundamental formula for arctan arises directly from the definition of the tangent in a right-angled triangle.
If:
tan(θ) = opposite / adjacent = x
Then the arctan formula is:
θ = arctan(x)
This formula helps you find the angle θ when you know the ratio 'x'. The output of the arctan function is the principal value, which is the angle within the restricted range of -90° to +90° (or -π/2 to π/2 in radians).
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| θ (theta) | The angle being calculated | Degrees or Radians | -90° to 90° or -π/2 to π/2 rad |
| x | The input value, representing the ratio of opposite to adjacent sides | Unitless | -∞ to +∞ |
Practical Examples
Understanding how to find arctan on a calculator is easier with examples.
Example 1: The Simplest Case
Imagine a right-angled triangle where the opposite side and the adjacent side are both equal, say 5 units each.
- Input (Ratio): opposite / adjacent = 5 / 5 = 1
- Calculation: θ = arctan(1)
- Result (Degrees): 45°
- Result (Radians): π/4 rad (approx. 0.785)
This shows that a 45° angle gives a tangent of 1. You can verify this with our Tangent Calculator.
Example 2: A Construction Scenario
A wheelchair ramp needs to be built. It rises 1 meter over a horizontal distance of 12 meters. What is the angle of inclination?
- Input (Ratio): Rise / Run = 1 / 12 ≈ 0.0833
- Calculation: θ = arctan(0.0833)
- Result (Degrees): ≈ 4.76°
The ramp makes an angle of about 4.76 degrees with the ground.
How to Use This Arctan Calculator
This tool makes it simple to find the inverse tangent. Here's a step-by-step guide on how to find arctan on this calculator:
- Enter the Value: Type the number (ratio) for which you want to find the arctan into the "Enter Value" field.
- Select Units: Choose whether you want the final angle to be in "Degrees" or "Radians" from the dropdown menu.
- Read the Result: The calculator instantly updates. The main result is shown in the large blue text. You'll also see the equivalent value in the other unit below it.
- Reset or Copy: Use the "Reset" button to return to the default value (1). Use the "Copy Results" button to save the output to your clipboard.
For finding arctan on a physical scientific calculator, you typically need to press the 'shift' or '2nd' key and then the 'tan' button.
Common Arctan Values
Here is a table of common arctan values for quick reference.
| Input (x) | arctan(x) in Degrees | arctan(x) in Radians |
|---|---|---|
| 0 | 0° | 0 |
| 1/√3 (≈ 0.577) | 30° | π/6 |
| 1 | 45° | π/4 |
| √3 (≈ 1.732) | 60° | π/3 |
Key Factors That Affect Arctan
- The Input Value: This is the most direct factor. As the input value increases, the resulting angle approaches 90° (or π/2 radians). As it decreases, it approaches -90°.
- Sign of the Input: A positive input value results in a positive angle (0° to 90°). A negative input value results in a negative angle (0° to -90°).
- Principal Value Range: The standard arctan function is defined to only return values between -90° and +90°. This is to ensure it remains a true function (one input gives one unique output).
- Unit of Measurement: Whether you work in degrees or radians changes the numerical representation of the angle, but not the angle itself. 180° is equivalent to π radians.
- Calculator Mode: When using a physical calculator, ensure it's set to DEG (degrees) or RAD (radians) mode to get the result in the desired unit. Our online tool handles this with the unit selector. For more details, see our guide on the Sine Calculator.
- Relationship to Atan2: For some applications, especially in computer programming, a two-argument function `atan2(y, x)` is used. It avoids division by zero and determines the correct quadrant for the angle, giving a full 360° range.
Frequently Asked Questions (FAQ)
Tan (tangent) takes an angle and gives a ratio. Arctan (inverse tangent) takes a ratio and gives an angle. They are inverse operations.
Usually, you press the 'SHIFT' or '2nd' key, then press the 'tan' key to access the tan-1 function. Then you enter your number and press '='.
As the input value 'x' approaches positive infinity, arctan(x) approaches 90° or π/2 radians. As 'x' approaches negative infinity, arctan(x) approaches -90° or -π/2 radians.
No, the standard arctan function's range is restricted to (-90°, 90°). To calculate angles in all four quadrants, you would use the atan2(y, x) function, which is not what this calculator does.
No, this is a common point of confusion. arctan(x) or tan-1(x) is the inverse function, while 1/tan(x) is the cotangent function, which is the reciprocal.
They are two different units for measuring angles. A full circle is 360 degrees or 2π radians. They represent the same angle, just like inches and centimeters represent the same length. Our Cosine Calculator also deals with these units.
The derivative of arctan(x) is 1 / (1 + x²).
The arctan of 0 is 0 degrees or 0 radians. This is because tan(0) = 0.