Arctan Calculator

The arctangent function, often written as arctan(x) or tan⁻¹(x), is the inverse of the tangent function in trigonometry. It is widely used in mathematics, physics, engineering, and computer graphics to determine angles when the tangent value is known.

Our Arctan Calculator simplifies this process by quickly computing the angle for any given tangent value. Whether you’re a student solving trigonometry problems, an engineer working with angles, or a programmer dealing with vector calculations, this tool provides accurate and instant results.

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What is Arctan?

• The arctangent function finds the angle whose tangent equals a given number.
• Mathematically:

θ=arctan⁡(x)if and only iftan⁡(θ)=x\theta = \arctan(x) \quad \text{if and only if} \quad \tan(\theta) = xθ=arctan(x)if and only iftan(θ)=x

• The result can be expressed in degrees or radians.

Example: arctan⁡(1)=45∘orπ4 radians\arctan(1) = 45^\circ \quad \text{or} \quad \frac{\pi}{4} \text{ radians}arctan(1)=45∘or4π​ radians

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How the Arctan Calculator Works

1. Input a Value: Enter the tangent value (x).
2. Choose the Unit: Select degrees or radians for the result.
3. Click Calculate: The calculator applies the inverse tangent function.
4. View the Result: Instantly get the angle corresponding to the input value.

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Step-by-Step Instructions

1. Open the Arctan Calculator.
2. Enter the tangent value (can be positive, negative, or zero).
3. Choose your preferred unit: degrees or radians.
4. Click Calculate.
5. Copy or note down the angle.

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Example Calculations

Example 1: Find the angle for tangent value 1. arctan⁡(1)=45∘orπ4 radians\arctan(1) = 45^\circ \quad \text{or} \quad \frac{\pi}{4} \text{ radians}arctan(1)=45∘or4π​ radians

Example 2: Find the angle for tangent value -√3. arctan⁡(−3)=−60∘or−π3 radians\arctan(-\sqrt{3}) = -60^\circ \quad \text{or} \quad -\frac{\pi}{3} \text{ radians}arctan(−3​)=−60∘or−3π​ radians

Example 3: Find the angle for tangent value 0. arctan⁡(0)=0∘or0 radians\arctan(0) = 0^\circ \quad \text{or} \quad 0 \text{ radians}arctan(0)=0∘or0 radians

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Benefits of Using the Arctan Calculator

✅ Fast & Accurate – No need for manual calculations.
✅ Supports Positive and Negative Values – Works for all tangent inputs.
✅ Flexible Units – Get results in degrees or radians.
✅ User-Friendly Interface – Simple input and instant output.
✅ Educational Tool – Helps students learn trigonometry and inverse functions.

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Applications of Arctan

1. Trigonometry: Solve triangles and find angles.
2. Physics: Determine angles in motion, vectors, and projectiles.
3. Engineering: Angle calculations in design, robotics, and construction.
4. Computer Graphics: Calculate rotation angles and orientations.
5. Navigation: Determine bearing and direction from slope ratios.

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Tips for Using Arctan

• Remember that arctan⁡(x)\arctan(x)arctan(x) outputs angles between −90∘-90^\circ−90∘ and 90∘90^\circ90∘ (or −π/2-\pi/2−π/2 to π/2\pi/2π/2 radians).
• If working in degrees, ensure your calculator is set correctly.
• For 2D vectors, use atan2(y, x) if you need the angle in all four quadrants.
• Negative inputs give negative angles; positive inputs give positive angles.
• Round results only after calculation for maximum accuracy.

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Frequently Asked Questions (FAQ)

Q1: What is arctan in simple terms?
A: Arctan finds the angle whose tangent equals a given number.

Q2: How is arctan different from tan?
A: Tan gives the ratio of opposite/adjacent for an angle. Arctan finds the angle for a given ratio.

Q3: Can arctan return negative angles?
A: Yes, for negative inputs, the result is a negative angle.

Q4: What units does arctan use?
A: Degrees or radians. You can choose the unit.

Q5: What is the range of arctan?
A: −90∘-90^\circ−90∘ to 90∘90^\circ90∘ or −π/2-\pi/2−π/2 to π/2\pi/2π/2 radians.

Q6: Can arctan be used in physics?
A: Yes, for vector angles, slopes, and projectile motion.

Q7: What if I input 0 into arctan?
A: The result is 0° or 0 radians.

Q8: Is arctan used in computer programming?
A: Yes, especially in graphics, robotics, and navigation.

Q9: Can arctan values exceed 90°?
A: No, standard arctan is limited to −90∘-90^\circ−90∘ to 90∘90^\circ90∘. Use atan2 for full-circle angles.

Q10: Can I calculate arctan for decimals?
A: Yes, any real number input is valid.

Q11: What is atan2?
A: atan2(y, x) finds the angle in all quadrants, accounting for both x and y signs.

Q12: Does arctan have a formula?
A: θ=arctan⁡(x)\theta = \arctan(x)θ=arctan(x) where tan⁡(θ)=x\tan(\theta) = xtan(θ)=x.

Q13: How do I convert radians to degrees?
A: Multiply by 180/π180/\pi180/π.

Q14: How do I convert degrees to radians?
A: Multiply by π/180\pi/180π/180.

Q15: Can arctan be used in 3D calculations?
A: Yes, often combined with atan2 for vector angles in 3D.

Q16: Is arctan only for positive values?
A: No, negative tangent values return negative angles.

Q17: Why is arctan important in trigonometry?
A: It helps solve right triangles when only tangent ratios are known.

Q18: Can arctan be approximated manually?
A: Yes, using series expansions or a scientific calculator.

Q19: Is the calculator free to use?
A: Yes, it provides instant and accurate results without any cost.

Q20: Can arctan help in navigation?
A: Yes, it calculates bearings, slopes, and angles of direction.

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Conclusion

The Arctan Calculator is an essential tool for students, engineers, scientists, and anyone working with angles or trigonometry. By entering a tangent value, you can quickly find the corresponding angle in degrees or radians without tedious calculations.