Is arctan the same as tan⁻¹?

Yes, arctan is the same as tan⁻¹ and both mean the inverse tangent function. However, tan⁻¹(x) is sometimes confused with (tan(x))⁻¹ = 1/tan(x) (cotangent), so arctan is the preferred notation to avoid ambiguity. On this calculator, arctan(x) gives the angle whose tangent is x.

Inverse Tangent

Q: What is inverse tangent (arctan)?

Inverse tangent (arctan or tan⁻¹) is the inverse function of tangent. Given a value x, arctan(x) returns the angle whose tangent equals x. For example, arctan(1) = 45° because tan(45°) = 1. The result always lies in the range (−90°, 90°) or (−π/2, π/2) radians.

Q: What is the domain and range of arctan?

The domain of arctan is all real numbers (−∞ to ∞). You can calculate the inverse tangent of any real number. The range (principal value) is (−90°, 90°) or (−π/2, π/2) in radians, exclusive. As x approaches ∞, arctan(x) approaches 90°, and as x approaches −∞, arctan(x) approaches −90°.

Q: What is arctan(1)?

Arctan(1) = 45° (π/4 radians). Since tan(45°) = 1, the inverse tangent of 1 is 45 degrees. This is one of the most commonly used values of arctan and appears frequently in trigonometry problems and geometry calculations.

Q: How is arctan different from cot?

Arctan (tan⁻¹) is the inverse of tangent: arctan(x) returns the angle whose tangent is x. Cotangent (cot) is the reciprocal: cot(x) = 1/tan(x). These are completely different functions. For example, arctan(1) = 45°, while cot(45°) = 1. The tan⁻¹ notation often causes this confusion.

Q: What are common arctan values?

Common arctan values include: arctan(0) = 0°, arctan(1/√3) = 30° (π/6), arctan(1) = 45° (π/4), arctan(√3) = 60° (π/3), arctan(∞) → 90° (π/2), arctan(−1) = −45° (−π/4), arctan(−√3) = −60° (−π/3). These correspond to standard angles on the unit circle.

Q: What is the derivative of arctan?

The derivative of arctan(x) is 1/(1 + x²). This derivative exists for all real numbers x. The integral of arctan(x) is x·arctan(x) − ½·ln|1 + x²| + C. These formulas are essential in calculus for solving integration and differentiation problems involving inverse trigonometric functions.

Calculate the inverse tangent (arctan/tan⁻¹) of any real number in degrees and radians. Free online arctan calculator for trigonometry, geometry, and calculus.

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Value (x):

About This Calculator

The inverse tangent calculator (also known as arctan or tan\u207b\u00b9 calculator) computes the angle whose tangent equals a given value. This free online tool is essential for students studying trigonometry, geometry, and calculus, as well as professionals in engineering, physics, architecture, and navigation who need to find angles from tangent ratios.

This calculator uses the mathematical formula arctan(x) = \u03b8 where tan(\u03b8) = x and \u03b8 is in the range (\u221290\u00b0, 90\u00b0) (the principal value range). Unlike arcsin and arccos which have a restricted domain, arctan accepts all real numbers from negative infinity to positive infinity. The result is computed using JavaScript's built-in Math.atan() function, ensuring IEEE 754 standard precision with results displayed to 4 decimal places in both degrees and radians.

Key Arctan Values Reference

x	arctan(x) in degrees	arctan(x) in radians
0	0\u00b0	0
1/\u221a3	30\u00b0	\u03c0/6
1	45\u00b0	\u03c0/4
\u221a3	60\u00b0	\u03c0/3
\u221e	90\u00b0	\u03c0/2
\u22121	\u221245\u00b0	\u2212\u03c0/4
\u2212\u221a3	\u221260\u00b0	\u2212\u03c0/3

Regional Notes

India (IN): Inverse tangent is taught in CBSE and state board Class 12 mathematics under inverse trigonometric functions. Students use arctan to solve trigonometric equations and right-triangle problems in competitive exams like JEE Advanced and JEE Mains.

United States (US): Arctan is introduced in high school precalculus and college trigonometry courses. US textbooks use tan\u207b\u00b9 notation alongside arctan. The function is available on all TI-84 and similar scientific calculators used in the US education system.

United Kingdom (UK): Arctan is covered in A-Level Pure Mathematics under trigonometry and functions modules. UK exam boards (AQA, Edexcel, OCR) emphasize understanding the restricted range and asymptotes of inverse trigonometric functions.

Frequently Asked Questions

What is inverse tangent (arctan)?

Inverse tangent (arctan or tan\u207b\u00b9) is the inverse function of tangent. Given a value x, arctan(x) returns the angle whose tangent equals x. For example, arctan(1) = 45\u00b0 because tan(45\u00b0) = 1. The result always lies in the range (\u221290\u00b0, 90\u00b0) or (\u2212\u03c0/2, \u03c0/2) radians.

What is the domain and range of arctan?

The domain of arctan is all real numbers (\u2212\u221e to \u221e). You can calculate the inverse tangent of any real number. The range (principal value) is (\u221290\u00b0, 90\u00b0) or (\u2212\u03c0/2, \u03c0/2) in radians, exclusive. As x approaches \u221e, arctan(x) approaches 90\u00b0, and as x approaches \u2212\u221e, arctan(x) approaches \u221290\u00b0.

Is arctan the same as tan\u207b\u00b9?

Yes, arctan is the same as tan\u207b\u00b9 and both mean the inverse tangent function. However, tan\u207b\u00b9(x) is sometimes confused with (tan(x))\u207b\u00b9 = 1/tan(x) (cotangent), so arctan is the preferred notation to avoid ambiguity. On this calculator, arctan(x) gives the angle whose tangent is x.

What is arctan(1)?

Arctan(1) = 45\u00b0 (\u03c0/4 radians). Since tan(45\u00b0) = 1, the inverse tangent of 1 is 45 degrees. This is one of the most commonly used values of arctan and appears frequently in trigonometry problems and geometry calculations.

How is arctan different from cot?

Arctan (tan\u207b\u00b9) is the inverse of tangent: arctan(x) returns the angle whose tangent is x. Cotangent (cot) is the reciprocal: cot(x) = 1/tan(x). These are completely different functions. For example, arctan(1) = 45\u00b0, while cot(45\u00b0) = 1. The tan\u207b\u00b9 notation often causes this confusion.

What are common arctan values?

Common arctan values include: arctan(0) = 0\u00b0, arctan(1/\u221a3) = 30\u00b0 (\u03c0/6), arctan(1) = 45\u00b0 (\u03c0/4), arctan(\u221a3) = 60\u00b0 (\u03c0/3), arctan(\u221e) \u2192 90\u00b0 (\u03c0/2), arctan(\u22121) = \u221245\u00b0 (\u2212\u03c0/4), arctan(\u2212\u221a3) = \u221260\u00b0 (\u2212\u03c0/3). These correspond to standard angles on the unit circle.

What is the derivative of arctan?

The derivative of arctan(x) is 1/(1 + x\u00b2). This derivative exists for all real numbers x. The integral of arctan(x) is x\u00b7arctan(x) \u2212 \u00bd\u00b7ln|1 + x\u00b2| + C. These formulas are essential in calculus for solving integration and differentiation problems involving inverse trigonometric functions.

How is inverse tangent used in real life?

Inverse tangent is used in navigation (calculating bearings from coordinates), computer graphics (rotation and 3D rendering), physics (finding angles from vector components), robotics (joint angle calculations), surveying (determining slope angles), and electrical engineering (phase angle calculations in AC circuits via the atan2 function).