Arcsin Calculator

What arcsin does, and the inputs it refuses

Sine takes an angle and returns a ratio: the length of the opposite side divided by the length of the hypotenuse. Arcsin runs that backwards. Hand it a ratio and it returns the angle whose sine matches, so arcsin(0.5) is 30 degrees, or 0.5236 rad, because the sine of 30 degrees is exactly one half. The NIST Digital Library of Mathematical Functions, section 4.23 defines the inverse circular functions and their principal values the same way, and the Wolfram MathWorld entry on the inverse sine collects the series and identities used later on this page.

The input is a sine, and a sine can never leave the interval from -1 to 1. The opposite side of a triangle cannot be longer than the hypotenuse it leans against, so any input beyond that interval has no real answer. arcsin(1.5) is not a large angle; it is a sign that the ratio was built from the wrong pair of sides, or that a digit slipped. The edge values themselves are fine: arcsin(-1) is -90 degrees, arcsin(0) is 0 and arcsin(1) is 90 degrees.

Notation trips people here. The label sin to the power minus 1, written with a small superscript, means the inverse function, not one divided by the sine. On calculators the same key is marked arcsin, asin or sin with a superscript, and the inverse sits above the sine key behind a shift or second function button.

Why the answer stops at 90 degrees

Sine repeats every 360 degrees and mirrors on the way, so one ratio belongs to many angles. A sine of 0.5 belongs to 30 degrees and to 150 degrees, and also to -210 and to 390. A function that returned all of them at once would not be usable, so arcsin is defined to return the single angle inside the closed interval from -90 to 90 degrees. That range is called the principal value.

The other answers are easy to recover once you have the principal one. For a positive sine, the second solution inside 0 to 180 is 180 minus the arcsin result, so 180 - 30 = 150 degrees shares the sine of 0.5. Beyond that, add or subtract full turns. For a negative sine the same rule applies on the lower half of the circle.

Which answer fits depends on the drawing. A swing at 30 degrees above the ground and a swing at 150 degrees are different physical positions that share a sine, so context, not the number, picks the branch. Navigation and robotics code usually reach for atan2 instead, because it takes both coordinate signs and returns the direction across all four quadrants. Arcsin cannot do that on its own: it sees one ratio and nothing else.

Degrees and radians, both rows at once

Every result on this page appears in both units. Multiply the radian value by 180 and divide by pi to get degrees, or multiply degrees by pi and divide by 180 to go back. The anchors are the sixth, quarter and half of pi: 30 degrees is 0.5236 rad, 45 degrees is 0.7854 rad and 90 degrees is 1.5708 rad.

Code only speaks radians. JavaScript's Math.asin, Python's math.asin and the C family's asin all return radians, so the degree line is a courtesy for the person reading the screen. The radian row also doubles as a magnitude check: an arcsin result in radians must sit between -1.5708 and 1.5708. Anything outside that window means the input was not a valid sine, or the mode switch on a handheld calculator is in the wrong place.

One subtlety near the ends of the range: the curve is steep at 0 and flattens hard near 1. Around zero, arcsin of a small value is close to the value itself, so arcsin(0.1) is 0.1002 rad, barely larger than the input. Near 1 the opposite happens, and tiny input changes swing the angle: arcsin(0.9) is 64.1581 degrees, arcsin(0.95) is 71.8051 degrees and arcsin(0.99) is 81.8904 degrees.

Right triangles: opposite side over hypotenuse

The triangle mode answers the question people bring to arcsin most often. In a right triangle, the sine of an angle is the opposite side divided by the hypotenuse, so the angle is arcsin of that ratio. A ramp that is 12 ft long along the incline and rises 1 ft at the top has a sine of 1 divided by 12, which is 0.0833, and the incline angle is arcsin of 0.0833, about 4.78 degrees.

The 3-4-5 triangle makes the idea visible. The side opposite the marked angle is 3 and the hypotenuse is 5, so the ratio is 0.6 and the angle is arcsin of 0.6, which is 36.8699 degrees. The same triangle also answers through the other ratios: the adjacent side over the hypotenuse is 0.8, so the arccos route gives 36.8699 degrees too, and the opposite over the adjacent is 0.75, so the arctan route lands on the same angle.

Picking the right inverse is the whole skill. Opposite and hypotenuse: arcsin. Adjacent and hypotenuse: arccos. Opposite and adjacent: arctan. Take the 3-4-5 triangle and feed the wrong pair to arcsin, say 3 and 4, and you get arcsin of 0.75, which is 48.5904 degrees instead of 36.8699. The wrong answer is close enough to look plausible, which is what makes the mistake expensive. When the two sides you have are the rise and the run, the arctan calculator is the tool; arcsin belongs to the cases where the slanted length is one of the two numbers.

Common whole-number triangles and ramp cases, all worked through the same ratio:

Snell's law and the critical angle

Optics leans on arcsin twice. Snell's law says that the index of the first medium times the sine of the incident angle equals the index of the second medium times the sine of the refracted angle. Solving for the ray inside the first medium gives arcsin of (index two times sine of theta two, divided by index one).

For a worked case, light travels in glass with index 1.5 and crosses into air after hitting the surface at 20 degrees from the normal. The sine of 20 degrees is 0.342, and 1.5 × 0.342 = 0.513, so the emerging ray sits at arcsin of 0.513, about 30.87 degrees. The ray bends away from the normal because air is the faster medium.

Push the incident angle higher and the refracted angle runs out of room: a sine cannot exceed 1. The angle where it runs out is the critical angle, arcsin of the second index divided by the first. For glass into air that is arcsin of 0.6667, which is 41.8103 degrees. Any ray hitting the surface more steeply than that undergoes total internal reflection and stays inside the glass, which is exactly how optical fiber guides light for kilometres and why a window seen from underwater turns into a mirror at shallow angles. For water into air the critical angle is arcsin of 0.7502, about 48.61 degrees. The critical angle only exists going from the denser medium to the lighter one; light entering glass from air always refracts, at every angle.

Critical angles into air, computed as arcsin of 1 over n from the indices shown (typical visible-light values, refractiveindex.info):

Values worth knowing

Four values cover most mental checks, and all four come from triangles with whole number sides or simple roots. A sine of 0.5 comes from half an equilateral triangle, 0.7071 from a right isosceles triangle, 0.8660 from the same equilateral half seen from the other vertex, and 1 from a degenerate triangle where the opposite side equals the hypotenuse, which is the 90 degree limit.

Two rows pair up around 45 degrees: a sine of 0.6 and a sine of 0.8 belong to the two acute angles of the same 3-4-5 triangle, and 36.8699 plus 53.1301 comes to 90. Spotting the partner value is a fast way to catch a typo, because a pair of angles from one triangle has to add to 90 degrees.

Checking an arcsin result without a calculator

The Taylor series for arcsin is x plus x cubed over 6, plus 3 x to the fifth over 40, plus 15 x to the seventh over 336, and onward. It converges fastest for small inputs, which covers the ratios that show up in ramps and refraction.

Try x = 0.5. The first term is 0.5, the second is 0.02083, and the third is 0.00234, so three terms give 0.5 + 0.02083 + 0.00234 = 0.52317 rad, which is 29.98 degrees against the true 30. Adding the fourth term, 0.00035, reaches 0.52352 rad, or 29.9958 degrees, an error of four thousandths of a degree.

Two identities extend the check. The first is arcsin(x) plus arccos(x) = 90 degrees for every valid input: arcsin(0.6) is 36.8699 degrees and arccos(0.6) is 53.1301 degrees. The second converts to the tangent, arcsin(x) = arctan of x over the root of (1 minus x squared). For x = 0.6 the root of (1 - 0.36) is 0.8, so the tangent route reads 0.6 divided by 0.8 = 0.75, and arctan of 0.75 is 36.8699 degrees, the same answer through a different door. A third check is the slope of the curve itself: the derivative is 1 over the root of (1 minus x squared), so at x = 0.6 the slope is 1 ÷ 0.8 = 1.25, meaning a change of 0.01 in the input moves the angle by roughly 0.7 degrees there.

Three errors that produce a wrong angle

The first error is an input outside -1 to 1. A calculator throws a domain error and spreadsheet software returns a #NUM! cell, but hand-built code often silently produces a NaN that propagates. When the ratio comes from measurements, an impossible value almost always means the two lengths were mislabelled, for example feeding the hypotenuse into the numerator.

The second is rounding the ratio before inverting it. The sine of 37 degrees is 0.6018, and feeding the rounded 0.6 back through arcsin returns 36.8699 degrees, off by more than a tenth of a degree. Near the steep end the damage grows: rounding 0.954 to 0.95 moves the answer from 72.6211 to 71.8051 degrees, a jump of eight tenths. Keep the ratio at full precision and round the final angle.

The third is trusting the principal value blindly. Arcsin answers within -90 to 90 degrees, but a sine of 0.5 also belongs to 150 degrees, and a physical setup aimed above 90 degrees from the reference direction will produce it. Surveyors and game developers handle this by keeping the drawing or the coordinate signs alongside the ratio. If the geometry says the angle is obtuse, take the arcsin result and subtract it from 180 degrees to land on the branch the drawing shows.