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Inverse Sine Calculator

Range: -1 – 1

Result

30.00 °

Angle

Angle in radians
0.5236 rad

An inverse sine calculator turns a sine value back into an angle. You give it a number between minus one and one; it tells you which angle has that number as its sine, in both degrees and radians. Everything it cannot do follows from one fact: a sine value does not belong to a single angle. A sine of 0.5 is the sine of thirty degrees, and also of a hundred and fifty degrees, and also of three hundred and ninety, and of every one of those minus a whole turn. The inverse sine function returns exactly one of them — the one sitting between minus ninety degrees and ninety degrees — and that choice is what the principal value means. It is not a claim that the others do not exist. The function is written arcsin, and the page reports its answer twice: once in degrees, once in radians, because both are asked for constantly and converting between them by hand is where most of the arithmetic errors in this area happen. Two limits bound the answer. The value has to lie between minus one and one, since the sine of an angle never leaves that band; ask for the inverse sine of 1.5 and there is no angle to give, so the page refuses instead of printing a blank. And the answer is signed: a negative input gives a negative angle rather than the equivalent angle in the lower half of the circle, which is the one place this page reads differently from the doubling and halving ones. The table below lists nine values with the principal angle, its radians, and the second angle in the circle that shares the same sine.

Nine sine values with the principal angle and the second solution

sin θθ (degrees)other θ (degrees)θ (radians)
-1-90270-1.5708
-0.866-60240-1.0471
-0.7071-45225-0.7854
-0.5-30210-0.5236
001800
0.5301500.5236
0.7071451350.7854
0.866601201.0471
190901.5708

The second column is what the function returns and the third is what it leaves out, and the pair is the whole point of the table. Read the middle rows first: a value of zero belongs to the pair zero and one hundred and eighty, and 0.5 — the value the page loads with — to the familiar pair thirty and one hundred and fifty. Down the whole table the two columns add up to one hundred and eighty, which is the arithmetic of the reflection, and the sum holds even at the ends. The two ends are where it stops being interesting, in two different ways. At the top the pair is ninety and ninety: the sine curve turns around at its peak, so the reflection of ninety degrees is ninety degrees again, and both columns print the same number. At the bottom the pair reads minus ninety and two hundred and seventy, which are the same direction written two ways — the second column keeps the sign the function returns and the third has already been placed on the circle, and the two only look like different angles because one of them is allowed to be negative. The fourth column is the second column converted; notice that it runs from minus 1.5708 to 1.5708, half of pi either way, while the angle columns run over a whole half turn.

Formula

θ = arcsin x x ∈ [−1, 1] θ ∈ [−90°, 90°] the other solution in [0°, 360°) is 180° − θ

x
The sine value you start with. It must lie between minus one and one: the sine of an angle cannot leave that band, so a value outside it corresponds to no angle at all
θ
The angle the function returns, called the principal value. It is always between minus ninety degrees and ninety degrees, and it is reported in degrees and in radians
180° − θ
The second angle in the circle with the same sine. Reflections across the vertical axis share a sine, which is why subtracting from a hundred and eighty finds the partner rather than adding to it

Reach for this page when you have a ratio and want the angle — a surveyed slope reported as a sine, a component of a force given as a fraction of the whole, the reading off a table where only the ratio was tabulated. The inverse sine function is also the one to reach for when a triangle problem hands you two sides and asks for the angle opposite one of them, since the sine of an angle is the opposite side over the hypotenuse. Two neighbours matter here. If you want the three ratios at an angle rather than the angle from a ratio, the trigonometry page runs the other way and the two agree wherever both are defined. And if what you actually have is the ratio of the adjacent side to the hypotenuse, that is the inverse cosine, whose range covers the upper half of the circle instead of the right half.

Worked examples

  1. A sine value of 0.5

    1. The value is inside the band from minus one to one, so there is an angle to find
    2. Thirty degrees is the angle in the principal range whose sine is 0.5, so the page reports thirty degrees
    3. In radians that is thirty times pi over a hundred and eighty, which is a sixth of pi, or 0.5236

    The value the page loads with, and the one that shows what the range costs. A hundred and fifty degrees also has a sine of 0.5 — it is the reflection of thirty degrees across the vertical axis — but the function returns one number and thirty is the one in range. The table makes the lost angle visible: 0.5 is the sixth of its nine rows, and the third column of that row is 150. Both angles are answers to the same question; only one of them is what a function is allowed to return.

  2. A sine value of 0.25, which is not a special angle

    1. A quarter is not the sine of any angle in the usual chart, so the answer will be a decimal
    2. The angle whose sine is 0.25 is 14.48 degrees, to two decimal places
    3. In radians the same angle is 0.2527, which is roughly a quarter of the sine value itself

    Most values land here rather than on a special angle, and this row is what a real lookup looks like. The radians column is worth a glance: for small angles the radian measure is very close to the sine value itself, which is why 0.2527 sits near 0.25. That is the small angle approximation, and the fact that it shows up unprompted in a table is the shortest demonstration of why radians are the unit that calculus uses.

  3. A negative sine value, minus 0.5

    1. Minus 0.5 is inside the band, so the page finds an angle
    2. The angle in the principal range whose sine is minus 0.5 is minus thirty degrees
    3. The radians follow the same sign: minus 0.5236

    The other place this page reads differently from the doubling and halving ones. There, a negative angle was brought into range before anything else happened and came back as a position in the upper half of the circle. Here the answer is left signed, because the principal range is the band from minus ninety to ninety and half of it is below zero. Thirty degrees below the horizontal and thirty degrees above it are the same angle with opposite signs, and the function reports the one that matches the input's sign. The table row for minus 0.5 shows both readings: minus thirty in the second column, and 210 in the third.

  4. A sine value of 1, at the edge of the band

    1. One is the largest sine any angle has, so the answer is the top of the principal range
    2. Ninety degrees has a sine of one, and no angle has a larger one
    3. In radians a right angle is half of pi, which is 1.5708

    The row where the two solutions collapse into one. Every other sine value in the band belongs to two different angles, but one belongs to ninety and to nothing else in the circle — the sine curve turns around there, so the reflection of ninety degrees across the vertical axis is ninety degrees again. The table shows it plainly: the second and third columns of the last row both read 90. That is not a duplicated entry, it is the curve flattening out at its peak.

Limitations

Three things this page does not do. First, a value outside the band from minus one to one has no answer and is refused rather than approximated — 1.5 is not the sine of anything. Second, the answer is the principal value only. Every sine value between minus one and one belongs to two angles in a full turn and to infinitely many across turns, and the page returns the one in the band from minus ninety to ninety; if the angle you want is the other one, the table's third column has it, and if it is the same angle plus or minus whole turns, nothing here will tell you how many. Third, the radians are printed to four decimals and the degrees to two, so a value that lands on a special angle after rounding will print as that angle — the inverse sine of 0.866 prints as sixty degrees, although the exact sine of sixty degrees is slightly larger than 0.866.

Frequently asked questions

Why does an inverse sine of 0.5 give 30 degrees and not 150?
Because both angles have a sine of 0.5, and a function can only return one number. The range of the inverse sine function is fixed as the band from minus ninety degrees to ninety degrees, and thirty is the angle in that band. A hundred and fifty degrees is a perfectly good answer to the equation, and the table on this page shows it in the third column, but it is not what the function returns.
What happens if I enter a value larger than 1?
The page refuses it, because no angle has a sine above one. The sine of an angle stays between minus one and one whatever the angle does, so the range of the inverse function is exactly that band and nothing outside it corresponds to an angle. This is different from a value like 1.5 in a division, where a large number simply produces a small one — here the question itself has no answer.
Why is the answer negative for a negative value?
Because the principal range runs from minus ninety degrees to ninety degrees, and half of it lies below zero. The sine function is odd — the sine of minus thirty is the negative of the sine of thirty — so undoing it on a negative value gives a negative angle. The equivalent angle in the lower half of the circle, 330 degrees for an input of minus 0.5, is the same direction; the page reports the signed one.
How do I find an angle from a sine value by hand?
Look up the value in a table of sines, or estimate from the band: a sine of 0.5 is thirty degrees, a sine of 0.7071 is forty-five, and a sine of 0.866 is sixty. For anything between those, a calculator is the practical route. The one thing to remember when doing it by hand is the second angle: every sine value in the open band belongs to two angles in a full turn, and a table usually lists only the first.
What is the inverse sine in radians?
The same angle, measured the other way. Thirty degrees is a sixth of pi, which is 0.5236 radians, and the page prints both so the conversion never has to be done by hand. Radians are the unit that makes the derivative of the sine equal to the cosine, which is why anything involving calculus states angles that way, while anything measured with a protractor states them in degrees.
Is the inverse sine the same as one divided by the sine?
No, and the two are easy to confuse because both are written with the word sine. The inverse sine undoes the sine: it takes 0.5 and returns thirty degrees. One divided by the sine takes 0.5 and returns 2, which is the cosecant of thirty degrees — a ratio, not an angle. The notation makes this worse, since the inverse is often written with a superscript minus one that looks exactly like an exponent.

References

Related calculators

Trigonometry CalculatorSine, cosine and tangent of any angle, in degrees or radians.Unit Circle CalculatorSine, cosine and tangent of an angle, read off the unit circle.Inverse Cosine CalculatorAngle from a cosine value, in degrees or radians.Inverse Tangent CalculatorAngle from a tangent value, in degrees or radians.