What an inverse trigonometric function does
An inverse trigonometric function runs the trigonometry backwards: you hand it a ratio of sides and it hands you the angle that produces that ratio. If you measure a ramp that rises 1 metre over a 12-metre run, the tangent of its inclination is 1/12 = 0.08333, and arctan(0.08333) = 4.7636° tells you the angle. Sine, cosine and tangent turn angles into ratios; arcsin, arccos and arctan turn ratios back into angles.
The complication is that no trigonometric function is one-to-one. Sine takes the value 0.5 at 30°, at 150°, at 390°, at −210°, and at infinitely many other angles spaced 360° apart. A function cannot return infinitely many answers, so mathematicians restrict the domain of the forward function to a stretch on which it climbs or falls without repeating, and define the inverse only on that stretch. The single answer that comes back is called the principal value.
Those restricted stretches are not arbitrary. For sine the choice is −90° to 90°, the interval containing zero on which sine rises steadily from −1 to 1. For cosine, zero is a maximum rather than a crossing, so the interval that works is 0° to 180°, on which cosine falls steadily from 1 to −1. For tangent it is −90° to 90° with the endpoints excluded, because tangent blows up there. Those three ranges are what the principal range output reports, and they are why arccos never returns a negative angle while arcsin frequently does. They are also the ranges fixed by ISO 80000-2, the international standard for mathematical signs and symbols, which is the convention this page follows for arcsin, arccos and arctan.
Domains, ranges and the reciprocal functions
The domain of an inverse function is the set of ratios it can accept, and it is inherited from the range of the forward function. A sine or cosine is a ratio of a leg to the hypotenuse, and the hypotenuse is always the longest side, so the ratio can never exceed 1 in magnitude. That makes −1 ≤ x ≤ 1 the domain of both arcsin and arccos; ask for arcsin(1.5) and there is no real angle to return, which is exactly what the calculator reports.
Tangent is a ratio of two legs, and either can be arbitrarily larger than the other, so arctan accepts every real number. Its output is squeezed into −90° to 90° and approaches those limits asymptotically: arctan(1000) = 89.9427° and arctan(1,000,000) = 89.99994°.
The three reciprocal inverses are defined through the first three. Since csc θ = 1/sin θ, the angle whose cosecant is x is the angle whose sine is 1/x, so arccsc x = arcsin(1/x). Likewise arcsec x = arccos(1/x). Both require a ratio of magnitude at least 1, because the reciprocal of something no bigger than 1 is never smaller than 1. Arccotangent is the odd one out: this calculator uses arccot x = π/2 − arctan x, giving a continuous function with range 0° to 180°, which is the convention in most calculus textbooks and the one that makes arccot(0) = 90°. Several programming environments instead compute arctan(1/x), which returns −45° where this page returns 135°. Check which convention your tool uses before comparing answers.
Worked example: the angle of a 7:12 roof
A rafter rises 7 inches for every 12 inches of horizontal run. Find the roof's angle above horizontal, then find every angle in one revolution with the same tangent.
- Identify the ratio. Rise over run is the opposite leg divided by the adjacent leg, which is the tangent: tan θ = 7/12 = 0.5833333.
- Apply the inverse. θ = arctan(0.5833333) = 0.528074 rad.
- Convert to degrees. 0.528074 × 180/π = 30.2564°. A 7:12 pitch is therefore just over 30 degrees.
- Find the second solution. The tangent repeats every 180°, so the other angle inside one revolution is 30.2564° + 180° = 210.2564°.
- Check by going forward. tan(30.2564°) = 0.583333, which is the ratio you started with, and tan(210.2564°) = 0.583333 as well.
Now try a sine instead. A 4-metre ladder reaches 3.6 metres up a wall, so sin θ = 3.6/4 = 0.9. Then θ = arcsin(0.9) = 64.1581°, and the second solution inside one revolution is 180° − 64.1581° = 115.8419°. The second answer is geometrically impossible for a ladder — an obtuse angle to the ground would put the top behind the wall — which is a reminder that the mathematics returns every solution and you must discard the ones your problem forbids.
Choosing the right answer from the list
Start from the principal value, then decide whether your physical situation permits the second solution. In a right triangle, every angle other than the right angle is acute, so only a principal value between 0° and 90° can be correct and the second solution is always spurious. That is why right triangle work almost never needs the full solution set.
In an oblique triangle solved with the law of sines, the second solution is often the real one. Given two sides and a non-included angle, both an acute and an obtuse triangle can fit the data — the classic ambiguous case — and the arcsine will only ever hand you the acute answer. You have to test the obtuse candidate yourself by checking that its angles still sum to 180°. The law of sines calculator flags that case explicitly.
In periodic problems — alternating-current phase, a rotating shaft, a tide model — all solutions matter, and you usually need the general form rather than the two inside one revolution: add 360°n to each listed answer, where n is any integer. For bearings and screen coordinates, prefer a two-argument arctangent, which uses the signs of both the numerator and the denominator to place the angle in the correct quadrant instead of collapsing that information into a single ratio.
Domain, range and behaviour of the six inverse functions
| Function | Accepts x in | Returns θ in | Value at x = 0 | Value at x = 1 |
|---|---|---|---|---|
| arcsin x | −1 to 1 | −90° to 90° | 0° | 90° |
| arccos x | −1 to 1 | 0° to 180° | 90° | 0° |
| arctan x | all real x | −90° to 90° (open) | 0° | 45° |
| arccsc x | magnitude at least 1 | −90° to 90°, excluding 0° | undefined | 90° |
| arcsec x | magnitude at least 1 | 0° to 180°, excluding 90° | undefined | 0° |
| arccot x | all real x | 0° to 180° (open) | 90° | 45° |
arcsin x + arccos x = 90° for every x in the domain, and arctan x + arccot x = 90° for every real x on this page's arccot convention. Both identities are quick checks on any answer.
Errors that turn a correct ratio into a wrong angle
- Reading sin⁻¹ as 1/sin. The superscript −1 on a function name means inverse, not reciprocal. sin⁻¹(0.5) = 30°, while 1/sin(0.5 rad) = 2.0858.
- Accepting the principal value in an ambiguous-case triangle. Given two sides and a non-included angle, check whether the obtuse partner also closes the triangle before discarding it.
- Using a one-argument arctangent for a bearing. The ratio −3/4 and the ratio 3/−4 are the same number but point in opposite directions; only a two-argument arctangent can tell them apart.
- Mixing arccot conventions. This page returns 135° for arccot(−1); a tool defined as arctan(1/x) returns −45°. They differ by exactly 180° for every negative input.
- Feeding a ratio outside the domain. An arcsin input above 1 usually means the opposite side and the hypotenuse were swapped, since a leg can never exceed the hypotenuse.
Key terms
- Principal value
- The single angle an inverse trigonometric function returns, chosen from the restricted interval on which the forward function is one-to-one.
- Ambiguous case
- A triangle specified by two sides and a non-included angle, where both an acute and an obtuse angle can satisfy the law of sines and both triangles may be valid.
- General solution
- The complete infinite family of angles sharing a given ratio, written by adding 360°n to each solution inside one revolution, or 180°n for tangent and cotangent.
Where inverse trigonometry shows up in practice
Surveying and construction use arctangent constantly, because grade, pitch and batter are all rise-over-run ratios waiting to be turned into angles. A 1:12 accessible ramp, the maximum running slope permitted by the 2010 ADA Standards for Accessible Design, is arctan(1/12) = 4.7636°. A 6:12 roof is arctan(0.5) = 26.5651°. Our slope calculator does the same conversion from two coordinates.
Machining and metrology use arcsine through the sine bar, a fixture with two rollers a known distance apart: stack gauge blocks of height h under one roller of a bar of length L and the bar tilts by arcsin(h/L). A 5-inch sine bar with a 2.5-inch stack sits at arcsin(0.5) = 30° exactly, which is why 5-inch and 10-inch bars are the standard sizes.
Navigation and graphics use the two-argument arctangent to recover a heading from displacement components, and physics uses arccosine to recover the angle between two vectors from their dot product. In each case the inverse function is the step that converts a measured ratio into a usable direction. When you need to go the other way, from a known angle to its ratios, use the sin cos tan calculator, and use the degrees to radians calculator when a library needs the angle in radians. For triangle geometry given three sides, the law of cosines calculator applies arccosine for you.