Arctan Calculator

What arctan answers, and the question it refuses

Tangent takes an angle and returns a ratio. Arctan runs the film backwards: give it the ratio of the opposite side to the adjacent side, and it returns the angle that produced it. A tangent of 0.5 belongs to an angle of 26.5651 degrees, so arctan(0.5) = 26.5651 degrees, or 0.4636 radians.

The catch is that tangent is not one to one. It repeats every 180 degrees, so 26.5651, 206.5651 and -153.4349 all have a tangent of 0.5. A function that returned all of them would not be usable, so arctan is defined to return the single value inside the open interval from -90 to 90 degrees. The NIST Digital Library of Mathematical Functions, section 4.23 calls this the principal value and gives the same range for the complex extension.

What arctan refuses to do is tell you which half of the circle you are in. It only sees a ratio, and a ratio throws away the signs of both legs. If your robot is heading toward the southwest, the ratio still looks like 1.3333, and arctan answers 53.13 degrees, which points northeast. The fix is atan2, covered below.

One boundary case is worth naming: arctan has no answer for a vertical line. As the ratio grows, the angle approaches 90 degrees but never arrives, which is why arctan(1000) is 89.9427 degrees rather than 90.

Degrees and radians describe the same angle

Every result on this page appears in both units, because the two are used by different trades. Degrees split a circle into 360 parts and are the default on drawing tools, phone levels and roof pitch charts. Radians measure the angle by the arc it cuts on a unit circle, so a full turn is 2 pi radians, a half turn is pi, and a right angle is pi ÷ 2.

The conversion is a single factor: multiply degrees by pi and divide by 180 to get radians. A 45 degree angle is 45 × pi ÷ 180 = 0.7854 rad. Going the other way, multiply radians by 180 and divide by pi.

Radians are not a fussier way of writing degrees; they are the unit that makes calculus come out clean. The derivative of arctan is 1 divided by 1 plus x squared, with no pi or 180 floating around, and at x = 1 that slope is 0.5. If the input were in degrees, that derivative would carry a factor of pi ÷ 180 and every physics formula would need the correction.

Since the derivative shrinks as x grows, arctan compresses large ratios into a narrow band. Between x = 10 and x = 100 the ratio changes tenfold, and the angle only moves from 84.2894 to 89.4271 degrees.

Slope, grade and roof pitch read as an angle

The most common use of arctan outside a classroom is turning a slope into an angle. Rise divided by run is a tangent, so the angle is arctan of that ratio, and the grade in percent is the same ratio multiplied by 100.

Take a roof with a 3 in 12 pitch: 3 ÷ 12 = 0.25, and arctan(0.25) = 14.0362 degrees, which is a 25 percent grade. The standard residential pitch of 4 in 12 gives 18.4349 degrees, and a steep 12 in 12 roof gives 45 degrees because the ratio is exactly 1.

A ramp is where the numbers carry legal weight. The 2010 ADA Standards set the maximum running slope for a new ramp at 1 to 12, which is 8.33 percent, and the U.S. Access Board guide to ramps states that limit with the recommendation to build flatter when space allows. In degrees that is arctan(1 ÷ 12) = 4.7636 degrees. A 1 to 8 slope, allowed only for a short rise on an existing site, is 7.125 degrees.

Percent grade and degrees diverge fast, which is why mixing them up produces bad ramps. A 25 percent grade sounds mild and is 14 degrees, steep enough that a wheelchair user has to work at it. The conversion is not proportional: 8.33 percent is 4.76 degrees and 100 percent is 45 degrees, not 57.

For a slope given in degrees and a known run, the rise is run times tan(angle). The roof pitch calculator takes that step for rafter lengths, and the slope calculator handles two points, grade and distance together.

Quadrants: why arctan alone points the wrong way

atan2 takes two arguments, y and x, instead of one ratio, and that extra sign is the whole point. It returns the angle of the point (x, y) measured anticlockwise from the positive x axis, in the range from -180 to 180 degrees. With both signs available it can place the point in the correct quadrant.

Navigation and game code rely on this. A bearing from one point to another is atan2(difference in north, difference in east) converted to degrees and then reduced to the range 0 to 360. Robotic arms, radar displays, joystick direction and camera aim all use the same call.

The argument order is the trap. JavaScript, Python, C and Java all take y first: Math.atan2(y, x). Excel takes x first: ATAN2(x_num, y_num), as Microsoft documents in the ATAN2 function reference. Swap the arguments and the answer mirrors across the diagonal, which looks plausible and is wrong by up to 90 degrees.

This page follows the programming convention and labels the boxes x and y, so a point at x = -3, y = 4 gives 126.87 degrees. The trigonometry calculator goes the other way, from an angle to the sine, cosine and tangent values, when the angle is what you already know.

Values worth remembering, and two you can derive

Three arctan values cover most quick estimates, and all three come from triangles whose sides are in simple ratios. A tangent of 1 comes from a right isosceles triangle, so arctan(1) is 45 degrees. A tangent of 1 ÷ root 3 comes from half an equilateral triangle, so arctan(0.5774) is 30 degrees. A tangent of root 3 comes from the same triangle the other way up, so arctan(1.7321) is 60 degrees.

Two results look surprising until you check them. arctan(1 ÷ 2) is 26.5651 degrees and arctan(1 ÷ 3) is 18.4349 degrees, and those two add to exactly 45 degrees. The identity behind it is arctan a + arctan b = arctan of (a + b) divided by (1 - ab), which for a = 0.5 and b = 0.3333 gives 0.8333 ÷ 0.8333 = 1, and arctan(1) is 45 degrees.

The other is the flatness at the top. arctan(10) is 84.2894 degrees and arctan(100) is 89.4271 degrees. A ratio ten times larger buys 5.1 degrees, and another factor of ten buys 0.5 degrees. Any measurement with a 5 percent error in that range produces an angle that is wrong by a small fraction of a degree, which is why steep slopes are better described by their ratio than by their angle.

Checking an arctan result without a calculator

The Taylor series for arctan is x minus x cubed over 3, plus x to the fifth over 5, minus x to the seventh over 7, and so on. It converges slowly near the ends of its range but is usable for values up to about 1, which makes it a decent way to check a result you doubt.

Try x = 0.5. The first term is 0.5, the second is 0.5 cubed over 3 = 0.0416667, and the third is 0.5 to the fifth over 5 = 0.00625. Summing the first three terms gives 0.4645833 rad, which is 26.6187 degrees against the true 26.5651, an error of 0.05 degrees. Adding the fourth term, 0.0011161, brings the estimate to 0.4634673 rad, or 26.5547 degrees, and the error drops to 0.01 degrees.

Two identities extend the check to larger values. arctan(x) = 90 degrees minus arctan(1 ÷ x) for positive x, so arctan(10) can be checked as 90 minus 5.7106, which is 84.2894 degrees. And arctan(x) = 2 × arctan of x divided by (1 + root of (1 + x squared)), which halves the argument at the cost of a square root.

A rough slope check works too: the derivative at x is 1 ÷ (1 + x squared), so near x = 0.5 the curve rises about 0.8 radians per unit of x. A change of 0.01 in the ratio should move the angle by about 0.46 degrees.

Small angles and steep faces: two ends where intuition slips

Near zero, arctan(x) is close to x in radians, and the two are often used interchangeably. At x = 0.1 the exact answer is 5.7106 degrees, while treating 0.1 as a radian gives 5.7296 degrees. The gap is 0.33 percent, which is fine for a sight line and too big for a survey.

The approximation improves as the angle shrinks. At x = 0.01 the error falls below 0.02 percent, which is why small angle formulas are standard in optics and astronomy: a one degree tilt is 0.01745 rad, and the tangent and the angle agree to four significant figures.

The other end is harder to work with. As x grows, arctan approaches 90 degrees and the sensitivity of the answer to the input collapses. Suppose a wall is measured 1000 mm out of plumb over a 10 m height: the ratio is 0.1 and the angle is 5.71 degrees. Now suppose the same wall is measured over only 1 m: the ratio is 1 and the angle is 45 degrees, even though the wall is the same. Slope angles are only meaningful together with the baseline they were measured on.

When the ratio becomes extreme, switch methods. If you know the opposite side and the hypotenuse, arcsin returns the angle directly and stays well conditioned near 90 degrees. If you know two sides of a triangle that is not right angled, the law of cosines avoids the problem altogether.

Calculator modes, rounding and three common errors

The first error is a calculator left in the wrong mode. Ask for arctan(1) in radian mode and you get 0.7854, which is correct and looks wrong to anyone expecting 45. The tell is the size of the number: angles in radians are always below 1.5708, so a result above that in radian mode means the inputs are wrong, and a result below 1.5708 in degree mode means you are looking at a nearly flat slope.

The second is using arctan where arcsin belongs. If the two numbers you have are the opposite side and the hypotenuse, their ratio is a sine, not a tangent. Feeding 3 and 5 into arctan gives 30.9638 degrees, and the correct answer from arcsin is 36.8699 degrees, a gap of almost 6 degrees.

The third is rounding the ratio before taking the angle. A slope of 0.0049 per metre rounds to 0.005, and arctan moves from 0.2807 to 0.2865 degrees, a 2 percent error at an angle where a tenth of a degree matters. Keep the ratio at full precision and round the angle at the end.

For output, two decimals is right for construction drawings, four for survey work and printed tables. Anything beyond four decimals claims precision the input rarely has, since a 1 percent error in the ratio already moves a 45 degree angle by about 0.3 degrees.