Arctan Calculator
Solve arctan problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations. Includes formulas and worked examples.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Arctan Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: arctan(x) = angle where tan(angle) = x | atan2(y, x) = full quadrant angle
Worked example — Ramp angle = 4.76 degrees | Slope = 8.33% | Meets ADA 1:12 requirement
Formula
arctan(x) = angle where tan(angle) = x | atan2(y, x) = full quadrant angle
The inverse tangent function accepts any real number and returns an angle between -90 and 90 degrees. The atan2(y, x) variant takes separate y and x values and returns the full angle from -180 to 180 degrees, correctly handling all four quadrants.
Worked Examples
Example 1: Converting a Slope to an Angle
Problem:A wheelchair ramp rises 1 meter over a horizontal distance of 12 meters. Find the ramp angle.
Solution:Slope ratio = rise / run = 1 / 12 = 0.08333 Angle = arctan(0.08333) Angle = 4.7636 degrees Angle in radians = 0.08314 radians Slope percentage = 0.08333 x 100 = 8.33% This meets ADA guidelines (max 1:12 slope = 8.33%) Verification: tan(4.7636 degrees) = 0.08333
Result:Ramp angle = 4.76 degrees | Slope = 8.33% | Meets ADA 1:12 requirement
Example 2: Finding Direction Angle with atan2
Problem:A game character at (0,0) needs to face a target at (-3, 4). Find the angle using atan2.
Solution:Using atan2(y, x) = atan2(4, -3) angle = atan2(4, -3) = 126.87 degrees angle in radians = 2.2143 This is in Quadrant II (x negative, y positive) Hypotenuse (distance) = sqrt(9 + 16) = sqrt(25) = 5 sin = 4/5 = 0.8, cos = -3/5 = -0.6 Note: arctan(4/-3) = arctan(-1.333) = -53.13 degrees (wrong quadrant!) atan2 correctly gives 126.87 degrees
Result:Direction = 126.87 degrees (Quadrant II) | Distance = 5 units
Frequently Asked Questions
What is arctan (inverse tangent)?
Arctan, also written as tan^(-1) or atan, is the inverse function of the tangent function. Given any real number x, arctan(x) returns the angle whose tangent equals x. Unlike arcsin and arccos which have restricted input domains, arctan accepts any real number from negative infinity to positive infinity. The output is restricted to the range (-90, 90) degrees or (-pi/2, pi/2) radians, which is the principal value range. For example, arctan(1) = 45 degrees because tan(45 degrees) = 1. The arctan function is one of the most widely used inverse trigonometric functions, appearing in navigation, physics, engineering, computer graphics, and complex number theory.
What is the difference between arctan and atan2?
While arctan(x) takes a single value (the tangent ratio y/x) and returns an angle between -90 and 90 degrees, atan2(y, x) takes two separate arguments (the y and x coordinates) and returns the full angle from -180 to 180 degrees. This distinction is crucial because arctan cannot distinguish between opposite directions. For example, arctan(1/1) and arctan(-1/-1) both equal 45 degrees, but the points (1,1) and (-1,-1) are in opposite directions. atan2(1,1) correctly returns 45 degrees while atan2(-1,-1) returns -135 degrees (or 225 degrees). The atan2 function properly handles all four quadrants and is the preferred function in programming and engineering applications where the full angle is needed.
What are the common arctan values?
The most important arctan values to memorize are: arctan(0) = 0 degrees, arctan(sqrt(3)/3) = arctan(1/sqrt(3)) = 30 degrees, arctan(1) = 45 degrees, arctan(sqrt(3)) = 60 degrees. For negative values: arctan(-1/sqrt(3)) = -30 degrees, arctan(-1) = -45 degrees, arctan(-sqrt(3)) = -60 degrees. As x approaches infinity, arctan(x) approaches 90 degrees, and as x approaches negative infinity, arctan(x) approaches -90 degrees. In decimal form: arctan(0.5774) is approximately 30 degrees, arctan(1.0) = 45 degrees exactly, and arctan(1.7321) is approximately 60 degrees. These values are derived from the special right triangles (30-60-90 and 45-45-90).
How is arctan used to calculate slopes and grades?
Arctan is the primary function for converting between slope ratios and angles. If a road rises 6 meters over a horizontal distance of 100 meters, the slope ratio is 6/100 = 0.06, and the angle of incline is arctan(0.06) = 3.43 degrees. Road grades are usually expressed as percentages: a 6% grade means the road rises 6 units per 100 horizontal units. To convert a grade percentage to an angle: angle = arctan(grade/100). Common examples: a 5% grade = arctan(0.05) = 2.86 degrees, a 10% grade = arctan(0.10) = 5.71 degrees, a 45-degree slope = arctan(1.0) = 100% grade. In construction, roof pitch is often specified as rise over run (like 4:12), and arctan converts this to the actual roof angle: arctan(4/12) = 18.43 degrees.
What is the derivative and integral of arctan?
The derivative of arctan(x) with respect to x is 1 / (1 + x^2). This elegant formula is always positive and decreasing, confirming that arctan is a strictly increasing function that becomes flatter as x gets larger. The derivative equals 1 at x = 0 and approaches 0 as x approaches positive or negative infinity. The integral of arctan(x) dx is x times arctan(x) - (1/2) times ln(1 + x^2) + C. Perhaps more importantly, the integral of 1/(1+x^2) dx = arctan(x) + C, which is one of the fundamental integral formulas in calculus. This integral appears in probability (the Cauchy distribution), physics (electric potential calculations), and signal processing (filter design).
How is arctan related to complex numbers?
In complex number theory, arctan connects to the complex logarithm through the formula arctan(x) = (1/(2i)) times ln((1+ix)/(1-ix)), where i is the imaginary unit. This relationship is used in complex analysis and has practical applications in electrical engineering. When representing a complex number z = a + bi in polar form, the argument (angle) is found using theta = atan2(b, a). For purely real complex numbers, this reduces to arctan. The arctan function also appears in the formula for pi: pi/4 = arctan(1), which leads to Leibniz formula: pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... Machin formula pi/4 = 4 arctan(1/5) - arctan(1/239) was historically used to compute digits of pi.
How is arctan used in computer graphics and game development?
In computer graphics, arctan (specifically atan2) is essential for calculating rotation angles, aiming directions, and camera orientations. When a game character needs to face a target, the angle is computed as atan2(targetY - characterY, targetX - characterX). In 2D games, this determines sprite rotation. In 3D graphics, arctan is used in converting between Cartesian and spherical coordinates for camera systems, environment mapping, and procedural textures. The function is also used in panoramic image stitching, lens distortion correction, and field-of-view calculations. Modern GPUs include hardware-accelerated atan2 instructions because the function is called millions of times per frame in rendering pipelines.
What is the connection between arctan and pi?
Arctan has a deep connection to the mathematical constant pi. The most direct relationship is arctan(1) = pi/4, which means four times the arctangent of 1 equals pi. This leads to several famous series and formulas for computing pi. The Gregory-Leibniz series states pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ..., which converges very slowly. Much faster is Machin formula from 1706: pi/4 = 4 times arctan(1/5) - arctan(1/239). Numerous similar Machin-like formulas exist, using different combinations of arctan values. Modern pi computation records have used variations of these arctan-based formulas along with fast algorithms for computing arctan of rational numbers. The BBP formula, discovered in 1995, allows computing individual hexadecimal digits of pi.
How does arctan behave for very large and very small inputs?
For very large positive inputs, arctan(x) approaches but never reaches 90 degrees (pi/2 radians). Similarly, for very large negative inputs, it approaches -90 degrees (-pi/2 radians). These horizontal asymptotes define the bounding behavior of the function. Specifically, arctan(10) = 84.29 degrees, arctan(100) = 89.43 degrees, and arctan(1000) = 89.94 degrees. For very small inputs near zero, arctan(x) is approximately equal to x (in radians). This small-angle approximation is accurate to within 1% for |x| less than 0.24 radians (about 14 degrees). More precisely, for small x: arctan(x) is approximately x - x^3/3 + x^5/5. This approximation is widely used in physics and engineering when dealing with small angles in optics, pendulum motion, and structural deflection.
What are common errors when using the arctan function?
The most critical error is using arctan(y/x) instead of atan2(y,x) when the full angle (all four quadrants) is needed. This mistake produces incorrect angles for points in quadrants II and III. Another common mistake is forgetting that arctan output is in radians by default in most programming languages, leading to confusion when degrees are expected. Division by zero is another pitfall: arctan(y/0) is undefined, but atan2(y, 0) correctly returns plus or minus 90 degrees. Students sometimes confuse arctan with the reciprocal 1/tan (cotangent), which is a completely different function. When computing slopes, confusing rise/run with run/rise produces the complementary angle instead of the correct one. Always verify results by checking that tan of the computed angle gives back the original input value.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Center of Mass Calculator
Calculate center of mass with inputs, formulas, and instant results.
🧮Centroid Calculator
Calculate centroid with inputs, formulas, and instant results.
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.
🧮Conic Sections Calculator
Calculate conic sections with inputs, formulas, and instant results.