Arctan Calculator
Solve arctan problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations. Includes formulas and worked examples.
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\nAngle = arctan(0.08333)\nAngle = 4.7636 degrees\nAngle in radians = 0.08314 radians\n\nSlope percentage = 0.08333 x 100 = 8.33%\nThis meets ADA guidelines (max 1:12 slope = 8.33%)\n\nVerification: 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)\nangle = atan2(4, -3) = 126.87 degrees\nangle in radians = 2.2143\n\nThis is in Quadrant II (x negative, y positive)\nHypotenuse (distance) = sqrt(9 + 16) = sqrt(25) = 5\nsin = 4/5 = 0.8, cos = -3/5 = -0.6\n\nNote: arctan(4/-3) = arctan(-1.333) = -53.13 degrees (wrong quadrant!)\natan2 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.