Arctan
Calculate arctan (tan⁻¹) of any real number with results in degrees and radians. Free online arctangent calculator with interactive graph for quick results.
About This Calculator
The arctan (inverse tangent or tan⁻¹) calculator computes the angle whose tangent equals a given real number. It is the inverse operation of the tangent function from trigonometry. For any real input x, arctan(x) returns an angle in the range (-90 deg, 90 deg) or (-pi/2, pi/2) radians.
The formula used is: y = arctan(x) where y is the output angle. Our calculator converts the radian result to degrees using the formula: degrees = radians x 180 / pi. Results are displayed in both units for your convenience.
The arctangent function has wide applications in mathematics, physics, engineering, and computer graphics. It is used to calculate angles from slope or gradient values, determine phase angles in signal processing, compute polar coordinates from Cartesian coordinates using atan2(y,x), and solve trigonometric equations.
Key properties of arctan include: arctan(-x) = -arctan(x) (odd function), arctan(x) + arctan(1/x) = pi/2 for x > 0, and the function approaches +/-90 deg asymptotically as x approaches +/-∞. The interactive graph displays this asymptotic behavior clearly.
Frequently Asked Questions
What is arctan (inverse tangent)?
Arctan (also written as tan⁻¹ or atan) is the inverse of the tangent function. It returns the angle whose tangent is a given number. For example, arctan(1) = 45 deg = pi/4 radians because tan(45 deg) = 1.
How do I use the arctan calculator?
Enter any real number in the Value (x) input field and click Calculate Arctan. The tool instantly displays the inverse tangent in both degrees and radians, along with a graph of the arctangent function curve showing how the output angle varies with the input.
What is the difference between arctan and cotangent?
Arctan (tan⁻¹) and cotangent (cot) are different functions. Arctan gives the angle whose tangent is x. Cotangent is the reciprocal of tangent: cot(x) = 1/tan(x). They are not the same, so avoid confusing tan⁻¹(x) with (tan(x))⁻¹.
What is the domain and range of arctan?
The domain of arctan(x) is all real numbers (-∞ to ∞). The range is (-pi/2, pi/2) in radians or (-90 deg, 90 deg) in degrees. This restricted range makes arctan a well-defined inverse function.
What is arctan(1) equal to?
arctan(1) = 45 deg = pi/4 radians. This is one of the most commonly used arctan values. In a right triangle with equal opposite and adjacent sides, the angle is exactly 45 deg.
What is the derivative of arctan(x)?
The derivative of arctan(x) is 1/(1 + x^2). This derivative exists for all real numbers and is positive everywhere, meaning the arctan function is strictly increasing across its entire domain.
What is the integral of arctan(x)?
The integral of arctan(x) is x·arctan(x) - 1/2·ln(1 + x^2) + C, where C is the integration constant. This can be derived using integration by parts, treating arctan(x) as 1·arctan(x).
What is arctan(0) in degrees and radians?
arctan(0) = 0 deg = 0 radians. Since tan(0) = 0, the inverse tangent of zero is also zero. This makes sense because the arctan function passes through the origin on its graph.