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Arctan Calculator

Q: What does the arctangent function actually compute?

Arctangent reverses the tangent operation: it takes a tangent ratio and returns the corresponding angle. In a right triangle, if you divide the opposite side by the adjacent side, arctan of that quotient gives you the angle. For instance, arctan(1) yields 45° because a tangent of 1 corresponds to a 45° angle. The function domain covers all real numbers, but its range is strictly between −90° and 90°.

Q: Is arctan the same as 1/tan?

No—this is a critical distinction. arctan (or tan⁻¹) denotes the inverse tangent function, which undoes the tangent operation. The expression 1/tan(x), by contrast, is the reciprocal of tangent, which equals cotangent. The inverse function and the reciprocal are entirely different operations. Always read tan⁻¹ as tan-inverse, not as (tangent)⁻¹.

Q: Why does arctangent never exceed ±90 degrees?

Arctangent is restricted to its principal value range (−π/2, π/2) to ensure it remains a true function—each input maps to exactly one output. The tangent function itself is periodic and unbounded in its full domain, so no inverse exists without restriction. By limiting the range, mathematicians created a well-defined, continuous inverse that covers all possible tangent ratios without ambiguity.

Q: How do you find arctan of a value like √3?

Recognize that tan(60°) = √3. Since arctangent undoes tangent, arctan(√3) = 60° or π/3 radians. Building familiarity with standard tangent values (0°, 30°, 45°, 60°, 90° and their radian equivalents) makes these lookups instant. A calculator confirms the result in milliseconds if you lack instant recall.

Q: Can arctangent handle negative numbers?

Yes—arctangent accepts all real numbers, including negative values. For example, arctan(−1) = −45° (or −π/4 radians). The function is odd, meaning arctan(−x) = −arctan(x). Negative inputs naturally yield negative angle outputs within the principal range, making arctangent symmetrical about the origin.

Q: What is the derivative of arctan used for?

The derivative d/dx(arctan(x)) = 1/(1 + x²) appears in optimization, rate-of-change problems, and integration. It quantifies how steeply the arctangent curve climbs at any given point. The denominator 1 + x² ensures the derivative is always positive, confirming that arctangent is strictly increasing everywhere. This property is essential in calculus for solving differential equations and analyzing motion or growth models.

The arctan function recovers the angle whose tangent is a given value, making it essential in trigonometry, navigation, and engineering. Unlike the forward tangent function, arctan accepts all real numbers as input and returns angles between −90° and 90° (or −π/2 to π/2 radians). Use this tool to convert tangent ratios back into angles, explore the function's mathematical properties, and reference exact values for common inputs.

Last updated: April 28, 2026

Creators Mateusz Mucha, BEng

Reviewers Anna Szczepanek and Jasmine J. Mah

8,107 people find this calculator helpful

Understanding Arctangent

Arctangent, denoted as arctan or tan⁻¹, is the inverse of the tangent function. Where tangent takes an angle and returns a ratio, arctangent does the reverse: it accepts a ratio and returns the corresponding angle.

In a right triangle, if you know the ratio of the opposite side to the adjacent side, arctan tells you the angle. For example, if this ratio equals 1, arctan yields 45°. The key distinction from the original tangent function is that arctangent has a restricted range: it always returns values between −π/2 and π/2 radians (−90° to 90°). This restriction, called the principal value, ensures that each input maps to exactly one output, making arctan a true inverse function.

The domain of arctangent is remarkably permissive—it accepts any real number, from negative infinity to positive infinity. This makes it more practical than some other inverse trigonometric functions that require bounded inputs.

Arctangent Formula

The arctangent function is defined as the inverse of the tangent function, restricted to the principal branch. Given a real number x, the arctangent returns the angle y such that the tangent of y equals x.

y = arctan(x) = tan⁻¹(x)

where −π/2 < y < π/2 and x ∈ ℝ

y — The angle in radians (or degrees) corresponding to the given tangent value.
x — The tangent value—any real number that serves as input to the arctan function.

Key Properties and Relationships

Arctangent interacts with other trigonometric functions in predictable ways. If you know arctan(x), you can derive sine and cosine values without computing the angle explicitly:

Sine: sin(arctan(x)) = x / √(1 + x²)
Cosine: cos(arctan(x)) = 1 / √(1 + x²)
Tangent: tan(arctan(x)) = x

Two additional identities simplify problem-solving: arctan(−x) = −arctan(x), meaning the function is odd; and arctan(x) = π/2 − arccot(x), linking arctangent to its inverse cotangent counterpart.

The derivative of arctan is elegant and useful in calculus: d/dx(arctan(x)) = 1/(1 + x²). This formula applies everywhere on the real line. The integral also has a clean form: ∫arctan(x) dx = x·arctan(x) − ½·ln(1 + x²) + C, where C is an arbitrary constant.

Common Arctangent Values

Several arctangent values appear frequently in mathematics and engineering:

arctan(0) = 0° (0 radians)
arctan(1) = 45° (π/4 radians)
arctan(√3) = 60° (π/3 radians)
arctan(1/√3) = 30° (π/6 radians)
arctan(−1) = −45° (−π/4 radians)

As x approaches positive infinity, arctan(x) approaches π/2; as x approaches negative infinity, it approaches −π/2. These asymptotic limits mean the arctangent curve flattens but never crosses these horizontal boundaries.

Practical Considerations

Keep these insights in mind when working with arctangent in real applications.

Distinguish tan⁻¹ from (tan)⁻¹ — The notation tan⁻¹(x) means arctan(x), not the reciprocal of tangent. The reciprocal 1/tan(x) is cotangent, written cot(x). Confusing these notations is a common source of error.
Range restrictions matter — Arctangent always returns angles in the range (−90°, 90°). If you need angles outside this range, you must add or subtract 180° appropriately. Calculators and programming languages often provide atan2(y, x) functions that handle quadrant selection automatically.
Numerical stability near boundaries — Near the asymptotes (x → ±∞), arctangent changes slowly. Small errors in x can produce negligible changes in the output. Conversely, near x = 0, arctangent is nearly linear, so precision is easier to maintain.
Unit consistency in mixed-unit problems — When applying arctangent to geometric or physics problems, verify that opposite and adjacent sides (or ratios) use the same units. Mixing units leads to nonsensical angle outputs.

Frequently Asked Questions

What does the arctangent function actually compute?

Arctangent reverses the tangent operation: it takes a tangent ratio and returns the corresponding angle. In a right triangle, if you divide the opposite side by the adjacent side, arctan of that quotient gives you the angle. For instance, arctan(1) yields 45° because a tangent of 1 corresponds to a 45° angle. The function domain covers all real numbers, but its range is strictly between −90° and 90°.

Is arctan the same as 1/tan?

No—this is a critical distinction. arctan (or tan⁻¹) denotes the inverse tangent function, which undoes the tangent operation. The expression 1/tan(x), by contrast, is the reciprocal of tangent, which equals cotangent. The inverse function and the reciprocal are entirely different operations. Always read tan⁻¹ as tan-inverse, not as (tangent)⁻¹.

Why does arctangent never exceed ±90 degrees?

Arctangent is restricted to its principal value range (−π/2, π/2) to ensure it remains a true function—each input maps to exactly one output. The tangent function itself is periodic and unbounded in its full domain, so no inverse exists without restriction. By limiting the range, mathematicians created a well-defined, continuous inverse that covers all possible tangent ratios without ambiguity.

How do you find arctan of a value like √3?

Recognize that tan(60°) = √3. Since arctangent undoes tangent, arctan(√3) = 60° or π/3 radians. Building familiarity with standard tangent values (0°, 30°, 45°, 60°, 90° and their radian equivalents) makes these lookups instant. A calculator confirms the result in milliseconds if you lack instant recall.

Can arctangent handle negative numbers?

Yes—arctangent accepts all real numbers, including negative values. For example, arctan(−1) = −45° (or −π/4 radians). The function is odd, meaning arctan(−x) = −arctan(x). Negative inputs naturally yield negative angle outputs within the principal range, making arctangent symmetrical about the origin.

What is the derivative of arctan used for?