Arctan Calculator (Inverse Tangent)
Calculate arctan(x) - the inverse tangent function
Arctan accepts any real number (no restrictions)
Common Arctan Values
arctan(0)
0°
0
arctan(√3/3 ≈ 0.577)
30°
π/6
arctan(1)
45°
π/4
arctan(√3 ≈ 1.732)
60°
π/3
arctan(-1)
-45°
-π/4
arctan(-√3/3)
-30°
-π/6
How to Use This Calculator
Enter Tangent Value
Input the tangent value (x) you want to find the angle for. Unlike arcsin and arccos, arctan accepts any real number.
Calculate
Click the "Calculate Arctan" button to get the angle in both degrees and radians.
Review Results
View the angle result and verify it by checking that tan(angle) equals your input value.
Formula
arctan(x) = θ, where tan(θ) = x
Principal value range: -90° to 90° (-π/2 to π/2 radians)
Definition:
The arctan function (also written as tan⁻¹) is the inverse of the tangent function. Given a tangent value x, arctan(x) returns the angle θ such that tan(θ) = x.
Domain and Range:
- Domain: All real numbers (-∞ to +∞) - no restrictions!
- Range: -90° < arctan(x) < 90° (-π/2 < arctan(x) < π/2 radians)
- The principal value is always in the first or fourth quadrant
Relationship to Tangent:
tan(arctan(x)) = x, for all real x
Key Properties:
- arctan(-x) = -arctan(x) (odd function)
- arctan(0) = 0°
- arctan(1) = 45°
- arctan(∞) = 90° (approaching)
- arctan(-∞) = -90° (approaching)
About Arctan Calculator
The Arctan Calculator (Inverse Tangent Calculator) is a powerful tool that finds the angle whose tangent equals a given value. Arctan is the inverse function of tangent, meaning it "undoes" the tangent operation. Unlike arcsin and arccos, arctan accepts any real number as input, making it particularly useful for slope calculations and angle determinations.
What is Arctan?
Arctan, written as arctan(x) or tan⁻¹(x), is the inverse trigonometric function of tangent. Given a tangent value x (any real number), arctan(x) returns the angle (in the principal range of -90° to 90°) whose tangent is x. It answers the question: "What angle has a tangent of x?"
When to Use This Calculator
- Slope Calculations: Find angles from rise/run ratios (slope = tan(angle))
- Triangle Problems: Find angles when you know opposite/adjacent side ratios
- Physics: Calculate angles in projectile motion, inclines, and forces
- Engineering: Determine angles for ramps, roofs, and mechanical systems
- Navigation: Calculate bearings and course angles
- Computer Graphics: Rotate objects and calculate angles between points
- Verification: Check solutions to trigonometric equations
Why Use Our Calculator?
- ✅ No Input Restrictions: Accepts any real number (unlike arcsin/arccos limited to [-1, 1])
- ✅ Instant Results: Get accurate angle values immediately in both degrees and radians
- ✅ Verification: Shows that tan(result) equals your input to confirm accuracy
- ✅ Easy to Use: Simple interface requiring only one input value
- ✅ 100% Free: No registration or payment required
- ✅ Mobile Friendly: Works perfectly on all devices
- ✅ Educational: Helps understand the relationship between tangent and arctan
Important Notes
- Unlimited Domain: Arctan accepts any real number - positive, negative, or zero
- Principal Value: The calculator returns the principal value between -90° and 90°
- Multiple Solutions: For any tangent value, there are infinitely many angles with that tangent (due to periodicity), but arctan returns only the principal value
- Quadrants: Arctan returns angles in quadrants I (0°-90°) and IV (-90° to 0°)
Frequently Asked Questions
What is arctan(1)?
arctan(1) = 45° (or π/4 radians). This is because tan(45°) = 1. At 45°, the opposite and adjacent sides of a right triangle are equal, so their ratio is 1.
What is arctan(0)?
arctan(0) = 0°. This is because tan(0°) = 0. At 0°, there is no vertical component, so the tangent ratio is 0.
Why can arctan accept any number, while arcsin and arccos are limited to [-1, 1]?
Tangent can produce any real number as output (from -∞ to +∞), while sine and cosine are limited to [-1, 1]. Since arctan is the inverse of tangent, it can accept all real numbers as input. The ratio of opposite/adjacent sides in a right triangle can be any value, making tangent's range unlimited.
What's the difference between arctan and tan⁻¹?
They are the same function! Both arctan(x) and tan⁻¹(x) represent the inverse tangent function. The notation tan⁻¹ is more common in some contexts, while arctan is clearer in distinguishing it from 1/tan(x), which is the reciprocal (cotangent), not the inverse.
How is arctan used to find angles from slopes?
Since slope = rise/run = tan(angle), we can use arctan to find the angle: angle = arctan(slope). For example, if a ramp has a slope of 0.1 (10% grade), the angle is arctan(0.1) ≈ 5.71°.
Can arctan return angles greater than 90°?
No, the principal value of arctan is always between -90° and 90° (-π/2 and π/2 radians), never reaching exactly ±90°. However, there are infinitely many angles with the same tangent value (like 135° has the same tangent as -45°), but arctan specifically returns the principal value in quadrants I or IV.