Arctan: Series, Identities and Exact Values

What arctan is, and what this page adds

Tangent takes an angle and hands back a ratio of two sides. Arctan runs that in reverse: hand it the ratio and it returns the angle between -90 and 90 degrees whose tangent is that ratio. That principal range exists because tangent repeats every 180 degrees, so the restriction is what makes the inverse a genuine function.

Most uses of arctan are one-liners: a ratio goes in, an angle comes out, and the arctan calculator covers that job along with slope angles and atan2 quadrant work. Spreadsheets expose the same pair as ATAN and ATAN2, which Microsoft documents for Excel. This page goes a level deeper and treats arctan as a function worth understanding on its own terms. It works through the complement identity that pairs arctan(2) with arctan(0.5), the Gregory series that turns arctan into pure addition and subtraction, the way John Machin exploited that series to compute 100 digits of pi in 1706, the addition rule for combining two angles, and the calculus that gives arctan its distinctive flattening curve.

Every claim here can be checked with the widget above: type a value and it shows the angle, the series partial sums as they close in on the true result, and the complement check that should land on exactly 90 degrees for positive inputs.

The complement identity: two arctans that always make 90 degrees

Draw a right triangle with a base of 1 and a rise of x. The angle at the foot of the triangle has an opposite side of x and an adjacent side of 1, so that angle is arctan(x). The angle at the top has its opposite and adjacent sides swapped, so it is arctan(1/x). Since the two acute angles of a right triangle always sum to 90 degrees, arctan(x) + arctan(1/x) = 90 degrees for every positive x. That is the whole proof, and the triangle in the figure above is it.

The numbers make the pairing concrete. arctan(2) = 63.4349 degrees, arctan(0.5) = 26.5651 degrees, and 63.4349 + 26.5651 = 90. The same holds for the less famous pair: arctan(1/3) = 18.4349 degrees and arctan(3) = 71.5651 degrees, which again add to 90. At x = 1 the identity folds onto itself: 45 + 45 = 90, so arctan(1) is its own complement and sits exactly halfway up the range.

The identity is also a working tool. It converts any large input into a small one: arctan of 4, 10 or 1,000 can be found by running arctan on 0.25, 0.1 or 0.001 and subtracting from 90 degrees. Negative x follows the same pattern with the sign flipped, pairing into -90 degrees, because arctan is an odd function and arctan(-x) = -arctan(x). For x = 0 the reciprocal does not exist, and arctan(0) = 0 degrees stands alone.

The Gregory series, one fraction at a time

Arctan has a power series that needs nothing but division and alternating signs. For arguments between -1 and 1, the NIST DLMF lists the expansion as arctan x = x - x cubed over 3 + x to the fifth over 5 - x to the seventh over 7, continuing forever. James Gregory published this form in the 1660s, and it turns angle-finding into arithmetic a patient person can do with pencil and paper.

Convergence is the catch. Each term shrinks by a factor of x squared and moves the sum the other way, so small inputs converge fast and inputs near 1 crawl. The table shows both facts at once for x = 0.5, in radians:

The true arctan(0.5) is 0.4636476 radians, or 26.5651 degrees, and twelve terms pin it down completely at this precision. The error after each step is bounded by the size of the next term, which is why you can stop the moment the next fraction is smaller than the accuracy you need. At x = 1 the same series turns into 1 - 1/3 + 1/5 = 0.8667 after three terms, still nowhere near its limit of pi/4 = 0.7854, and hundreds of terms are needed for even a few correct digits. The widget above runs the series live and reports the gap at 1, 2, 3, 5, 8 and 12 terms, folding inputs outside the window back with the complement identity first.

Machin's formula: 100 digits of pi from two small arctans

In 1706 John Machin, a professor of astronomy in London, combined two fast-converging arctans into what is still one of the most elegant formulas in recreational mathematics: pi over 4 equals 4 times arctan(1/5) minus arctan(1/239). He used it to reach 100 decimal places of pi, smashing the record of the day, and the identity holds exactly, not approximately.

The numbers show why it converges so much faster than the plain series at x = 1. arctan(1/5) = 0.1973956 radians, so 4 x 0.1973956 = 0.7895824. arctan(1/239) = 0.0041841 radians, a contribution so small it barely registers. Subtracting gives 0.7895824 - 0.0041841 = 0.7853983, and pi/4 = 0.7854 to the same precision; the single final digit of difference comes from carrying only seven decimals in the inputs.

The deep reason is the error term. Each term of the Gregory series shrinks by the square of the argument, so working at 1/5 shrinks errors 25-fold per term and working at 1/239 shrinks them 57,121-fold per term. Where the naive series needs hundreds of terms for a handful of digits at x = 1, Machin's pairing gets 100 digits with about 70 terms of arctan(1/5) and barely one or two of arctan(1/239). Modern record hunters kept the same trick with ever-larger denominators; the formula is a lesson in choosing your argument before grinding terms.

Adding two angles with the addition rule

The tangent addition formula rearranges into a rule for combining arctans. For a and b with a x b below 1, arctan a + arctan b = arctan of (a + b) over (1 - ab). In words: add the ratios for the numerator, multiply them for the correction in the denominator, and one arctan reabsorbs the pair.

A worked pair makes it concrete. Take arctan(0.5) plus arctan(0.25). The combined tangent is (0.5 + 0.25) / (1 - 0.125), which comes to exactly 6/7, and arctan(6/7) = 40.6013 degrees. You can confirm on the widget that this equals the sum of 26.5651 and 14.0362, the two separate angles.

The guard on a x b matters. When the product reaches 1 the denominator hits zero, which is the signature of the sum landing on 90 degrees, where tangent has no finite value. When the product exceeds 1, the combined angle crosses into the next quadrant and the raw formula lands 180 degrees away from the truth, so you add or subtract 180 by hand. The cases:

Checking that last row: arctan(2) = 63.4349 degrees and arctan(3) = 71.5651 degrees, and 63.4349 + 71.5651 = 135. The formula alone reports arctan of -1, which is -45 degrees, exactly one half turn below the true 135, and the correction rule fixes it.

Calculus: slope 1 over (1 + x squared) and the area behind it

Arctan earns its place in calculus through its derivative, which the DLMF tabulates as 1 / (1 + x squared). The slope is steepest at x = 0, where it equals 1, and it decays as x grows: at x = 0.5 the slope is 1 / (1 + 0.25) = 0.8, and by x = 10 it is below 0.01. That decay is the curve visibly flattening toward its 90 degree ceiling without ever reaching it, the same behavior that keeps the principal range open.

Integrating that derivative recovers arctan, and integrating arctan itself produces a surprise: a logarithm. The antiderivative is x times arctan(x) minus half of ln(1 + x squared). At x = 1 the pieces are arctan(1) = 0.7854 and half of ln(2) = 0.3466, so the definite area from 0 to 1 is 1 x 0.7854 - 0.3466 = 0.4388. An inverse trig function hiding a natural log is the standard warning that the family of elementary functions is more tangled than it looks.

Two consequences follow for anyone using arctan numerically. First, the small slope for large x means arctan is extremely insensitive to changes in a big input: arctan(100) and arctan(101) differ by less than a ten-thousandth of a degree, which is why atan2 exists to recover direction from two coordinates instead of one lossy ratio. Second, the derivative being a rational function makes arctan the natural antiderivative for expressions like 1 over (1 + x squared), which is why it appears in partial fraction work and in the normal distribution's cumulative tables.

Exact values worth memorizing, and how to derive the rest

Only a handful of arctan values come out clean, because clean angles produce clean tangent ratios. Memorize the half dozen below and derive everything else from the identities on this page.

Everything else comes from the two moves this page has covered. The complement identity converts any value above 1 into one below it: arctan(12) is 90 degrees minus arctan(1/12). The addition rule builds new angles from known ones, which is exactly how Machin constructed his formula and how the half-angle values fall out too. For example, arctan(1/3) = 18.4349 degrees and arctan(3) = 71.5651 degrees form a complementary pair, and neither needed a new triangle, just the 1-and-x triangle with its legs relabeled.

A practical habit: when an arctan shows up in a problem, first ask whether the argument is near one of the five table values or the reciprocal of one. Estimating against 0.5774, 1 or 1.7321 usually gets you within a degree or two before any series work begins, and the widget above closes the remaining gap term by term.