One angle out of infinitely many — and all the others too.
The angle, and every other one
A calculator gives you one of infinitely many.
asin(0.5)
30°
The principal value, restricted to −90° to 90°. Every angle with this sine is 30° + 360n, and also 150° + 360n.
Principal value
30°
restricted to −90° to 90°
In radians
0.52359878
degrees × π/180
The other one in a turn
150°
equally valid, and a calculator hides it
Exact angle?
30°
one of the special angles
The general solution
30° + 360n, and also 150° + 360n
n is any whole number. A calculator returns one member of this set; which member your problem wants is decided by the context, not by the arithmetic.
The answer is restricted to −90° to 90°. Without that restriction the inverse would not be a function at all — sine takes every value in its range infinitely often, so "the" angle is a choice rather than a fact.
The full set is 30° + 360n, and also 150° + 360n. A calculator gives you one member of it, and in a real problem the context picks which.
This restriction is exactly why the SSA triangle case is ambiguous: asin hands back the acute angle, and the obtuse one with the same sine is equally valid.
A value outside −1 to 1 is refused with the reason, because no angle has that sine or cosine at all.
What this tool shows
arcsin 0.5 is 30°. It is also 150°, and 390°, and −210°. A calculator returns one member of that family because the restriction is what makes the inverse a function at all — and which member your problem wants is decided by the context, not the arithmetic.
Arcsin, arccos and arctan of any value
The principal value and its range
The other solution within one turn
The general solution, for every n
Which values have no inverse at all
Why atan2 exists
All three inverses The general solution The range explained Out-of-range refused
Updated 7 September 2026 · Works in any browser, no installation
A calculator gives you the principal value, which is one of infinitely many angles. arcsin 0.5 = 30°, and 150° has the same sine, and so does every one of those plus a whole number of turns. The restriction to a fixed window is what makes the inverse a function; it is not the calculator being unhelpful.
At a glance
Formula shown
arcsin returns [\u221290\u00b0, 90\u00b0], arccos returns [0\u00b0, 180\u00b0], arctan returns (\u221290\u00b0, 90\u00b0). The general solutions are \u03b8 + 360n and 180\u00b0 \u2212 \u03b8 + 360n for sine, \u00b1\u03b8 + 360n for cosine, and \u03b8 + 180n for tangent.
Scenario support
Recovering an angle from a ratio; solving a trigonometric equation where every solution matters; understanding why a triangle solver returned the angle it did.
Educational estimate
Planning support from the values you enter — not professional advice.
Why the range is restricted
A function has to give one output for each input. Sine does not: sin 30° and sin 150° are both 0.5, and so are infinitely many others.
So “the angle whose sine is 0.5” is not a well-posed question. To define an inverse at all, you have to restrict sine to a stretch where it takes each value once — and the conventional stretch is −90° to 90°, where it rises steadily from −1 to 1.
Inside that window the inverse exists and is unique. Outside it, the information is simply gone: the value 0.5 does not remember whether it came from 30° or 150°.
This is not a quirk. It is the reason the SSA triangle case is ambiguous, and the reason a trigonometric equation has infinitely many solutions while a calculator shows one.
The three ranges, and why they differ
arcsin: −90° to 90°. Sine rises steadily across that stretch, from −1 to 1, hitting every value once.
arccos: 0° to 180°. Cosine falls steadily there, from 1 to −1. It cannot use the same window as sine, because cosine is symmetric about 0° — cos(−30°) and cos 30° are equal, so −90° to 90° would fail the one-output rule.
arctan: −90° to 90°, open at both ends. Tangent runs from −∞ to ∞ across that stretch, and the endpoints are excluded because tangent is undefined there.
The different windows catch people out. arcsin and arctan can return negative angles; arccos never does. That asymmetry is a consequence of the shapes of the functions, not a convention someone chose freely.
The general solution
For an equation rather than a lookup, the family is the answer.
sin θ = k: θ = arcsin k + 360n, and also 180° − arcsin k + 360n. Two branches, because sine is symmetric about 90°.
cos θ = k: θ = ±arccos k + 360n. One expression covering both branches, because cosine is symmetric about 0°.
tan θ = k: θ = arctan k + 180n. Only one branch, because tangent already repeats every 180°.
Which member you want comes from the problem. An angle in a triangle is between 0° and 180°; a bearing is between 0° and 360°; a phase can be anything. The page gives the whole set and leaves that choice where it belongs.
Out of range
Sine and cosine never leave −1 to 1, so arcsin 1.5 has no answer. Not a very large angle — no angle at all.
When this comes up in a real problem it is usually diagnostic. In the sine rule, asking for the arcsine of something above 1 means the sides and angle you supplied describe no triangle: the side is too short to reach.
The page refuses with that explanation rather than returning NaN, because the refusal is the useful output.
Arctangent has no such limit. Tangent takes every real value, so arctan accepts anything — and approaches ±90° without ever reaching them.
Why atan2 exists
arctan takes a ratio, and a ratio has lost information that a point still has.
The point (1, 1) and the point (−1, −1) both give y/x = 1, so arctan returns 45° for both. But they are 180° apart, in opposite quadrants. The ratio cannot tell them apart because the two minus signs cancelled.
atan2(y, x) takes the two coordinates separately, so it can use their signs to pick the right quadrant. It returns a full −180° to 180°, and it handles x = 0 without dividing by it.
Any time you are converting a point to an angle — a heading, a direction, a vector’s argument — atan2 is the one to use. Reaching for atan(y/x) is a bug that works correctly in two quadrants out of four.
Notation
sin⁻¹x means arcsin x, the inverse function. It does NOT mean 1/sin x, which is cosecant.
This is a genuine inconsistency in mathematical notation, not a misunderstanding: sin²x DOES mean (sin x)², so the superscript means one thing for 2 and another for −1.
“arcsin” is unambiguous and is preferred in writing for that reason. The name comes from the unit circle, where the angle in radians is the length of the ARC it subtends — so arcsin is literally “the arc whose sine is”.
Sources and methodology
The principal-value conventions are standardised; these are the references.
Method. Each inverse reports its own range and the general solution derived from the symmetry of its function — the second in-turn solution for sine is 180° − θ, for cosine it is −θ, and for tangent it is θ + 180°. Values outside −1 to 1 are refused for sine and cosine with the reason, since no angle has such a sine at all. The suite checks that every returned value lies inside its stated range and round-trips through the forward function. That engine is verified on every change against 71 hand-written assertions, including that arcsin and arccos always land inside their stated ranges and round-trip through sine and cosine across two hundred generated values. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Trigonometric FunctionsAll six functions at once with exact surd values at the sixteen special angles — and tan 90° reported as undefined rather than as the 1.6 × 10¹⁶ a double returns.
Unit CircleWhere an angle lands, with exact surd coordinates and the radian measure as a multiple of π — because the coordinates ARE the cosine and sine.
Reference AngleThe acute angle to the x-axis — never the y-axis — with the quadrant rule written out using your own numbers and the sign of each trig function beside it.
TriangleSolve any triangle from three measurements — and when two sides and a non-included angle describe TWO triangles, this one shows both instead of picking one.
Angle Between Two VectorsDot product, magnitudes and the angle as separate steps, in the plane or in space — with exactly parallel vectors returning 0° rather than a floating-point smudge.
Sum and Difference IdentitiesEach identity evaluated twice — through the formula and by computing the combined angle directly — so the agreement between them is evidence rather than a restatement.
The forward direction — an angle to its six values, with exact surds — is the Trigonometric Functions Calculator, and the Unit Circle Calculator shows where both directions come from.
Educational use disclaimer
This is an educational tool. The principal value is what a calculator returns; whether it is the angle your problem wants is a question the arithmetic cannot answer, and the page says so.
Published the inverse trig page reporting the general solution as well as the principal value, since a calculator returns one member of an infinite family and which member is right is a question the arithmetic cannot answer.
Explains the restricted range as the reason the inverse is a function at all, and connects it directly to why the SSA triangle case is ambiguous.
A value outside −1 to 1 is refused with the reason rather than returning NaN — in a triangle problem it usually means the measurements describe no triangle.
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