Introduction
This arcsin calculator finds the angle when you know the sine value. Arcsin is also called inverse sine, and you may see it written as arcsin(x) or sin−1(x). Type a number between −1 and 1, and the tool gives you the angle right away.
You get the answer in three units at once: degrees, radians, and gradians. When the answer is a neat multiple of π, like π/6, the exact form is shown too. A step-by-step solution explains how the answer was found, and a graph of y = arcsin(x) marks your point on the curve, much like you would see in a graphing calculator.
You can enter your value in a few ways:
- Decimals, like 0.5 or −0.333
- Fractions, like 1/2 or −3/4 (see our fraction calculator for help with fractions)
- Square roots, like sqrt(3)/2 (our square root calculator can work these out)
- Or just drag the slider
Remember, sine values only go from −1 to 1, so arcsin only works with numbers in that range. The answer is always the principal value, an angle from −90° to 90° (−π/2 to π/2 radians). This makes the calculator handy for trigonometry homework, geometry, physics, and any job that uses angles. For the forward functions, try the trig calculator or the cos calculator, and for the inverse tangent see the tan inverse calculator.
How to use our Arcsin Calculator
Type a sine value between −1 and 1, and the arcsin calculator gives you the angle in degrees, radians, and gradians, plus an exact π form, a step-by-step solution, and a graph of y = arcsin(x).
Value of x (the sine value): Enter the number you want the arcsine of. You can use a decimal like 0.5, a fraction like 1/2, or a root like sqrt(3)/2. The value must stay between −1 and 1, or arcsin is not defined.
Or drag to set x: Move the slider to pick a value of x from −1 to 1. The input box and all results update as you drag.
Decimal places: Choose how many decimals to show in your answer: 2, 4, 6, or 8. Pick more decimals for more exact results. If you need to tidy up a long answer, the rounding calculator and the sig fig calculator can help.
Graph angle unit: Pick degrees, radians, or gradians for the y-axis of the graph. Your point on the curve is marked in red.
Calculate and Reset: Click Calculate to see the answer, or press Enter. Click Reset to go back to the default value of 0.5.
What Is Arcsine (arcsin)?
Arcsine is the inverse of sine. Sine takes an angle and gives you a number. Arcsine does the opposite: you give it a number, and it gives you back the angle. You may also see it written as sin−1(x) or asin(x).
For example, sin(30°) = 0.5, so arcsin(0.5) = 30°. The two functions undo each other.
The Input Must Be Between −1 and 1
Sine never gives an answer smaller than −1 or bigger than 1. Because of that, arcsine only works for numbers from −1 to 1. This is called the domain. If you type in 2 or −5, there is no answer, and the calculator will tell you it is undefined.
The Answer Has Limits Too
Many angles share the same sine value. For example, sin(30°) and sin(150°) both equal 0.5. To keep one clear answer, arcsine only returns angles from −90° to 90° (that is −π/2 to π/2 radians). This one answer is called the principal value. If you need to work with other angles in the same family, the angle calculator is a useful companion.
Angle Units
- Degrees (°) — a full circle is 360°. Most common in school and building work.
- Radians (rad) — a full circle is 2π. Used in higher math, physics, and computer code. Radians are also what the arc length calculator and the circle calculator use.
- Gradians (grad) — a full circle is 400 grad. Used in some surveying jobs.
To change radians into degrees, multiply by 180/π. To change degrees into gradians, multiply by 10/9.
Common Arcsine Values
| x | arcsin(x) in degrees | arcsin(x) in radians |
|---|---|---|
| −1 | −90° | −π/2 |
| −0.5 | −30° | −π/6 |
| 0 | 0° | 0 |
| 0.5 | 30° | π/6 |
| √2/2 | 45° | π/4 |
| √3/2 | 60° | π/3 |
| 1 | 90° | π/2 |
Where Arcsine Is Used
In a right triangle, sine equals the opposite side divided by the hypotenuse. So if you know those two sides, arcsine finds the missing angle. Pair this tool with the right triangle calculator, the hypotenuse calculator, or the Pythagorean theorem calculator when you need the sides first, and use the triangle angle calculator, law of sines calculator, or law of cosines calculator for triangles that are not right-angled.
People use this in construction to find roof slopes and ramp angles — see the roof pitch calculator, the ramp slope calculator, and the slope calculator — in navigation to find headings, in physics for light bending (Snell's law) and for wave motion, and in games and graphics to point objects in the right direction.
A Quick Way to Check Your Answer
Take the angle you got and find its sine. You should get your starting number back. If arcsin(0.5) = 30°, then sin(30°) should equal 0.5. If it does, your answer is right. You can run that check in the trig calculator, and if your result is a little off, the percent error calculator will show how far off it is.