What is arcsine
Arcsine (written arcsin or sin⁻¹) is the inverse of sine: it answers "what angle has this sine?" Since sin(30°) = 0.5, arcsin(0.5) = 30°. The output ranges from −90° to 90°.
Sine takes an angle and returns a ratio; arcsine takes that ratio and returns the angle. In a right triangle the sine of an angle is the opposite side divided by the hypotenuse, so if you know those two sides you can divide them and feed the ratio in here to recover the angle itself. For a ratio of opposite to adjacent rather than opposite to hypotenuse, the inverse you want is the arctangent calculator.
How to use this calculator
Enter a value between −1 and 1 and read the angle in degrees. Values outside that range have no real arcsine and are rejected. To go the other way, use the sine calculator.
A worked case: a ramp rises 1 m over a 2 m sloped length. The sine of the incline is 1 ÷ 2 = 0.5, so the angle is arcsin(0.5) = 30°. Note that the input is a plain ratio, not a length — dividing the two sides cancels the units before you type anything in.
Why the input is limited
Because sine never exceeds 1 or drops below −1, only inputs in that interval have an angle. The calculator returns the principal value between −90° and 90°.
The restriction to a principal value exists because sine repeats: 30°, 150°, 390° and endlessly many other angles all have a sine of 0.5. To be a function, arcsine has to pick one, and by convention it picks the angle nearest zero. If your problem involves an obtuse angle, take 180° minus the principal value — a common oversight when solving triangles with the law of sines.
Frequently asked questions
- What is arcsin(0.5)?
- arcsin(0.5) = 30°, the principal angle whose sine is 0.5.
- Why must the input be between −1 and 1?
- Sine never goes outside −1 to 1, so values beyond that range have no real arcsine.
- What is the output range of arcsine?
- The principal value of arcsine ranges from −90° to 90°.
- Is arcsine the same as 1 ÷ sine?
- No. Arcsine is the inverse function of sine, not its reciprocal (which is cosecant).
- How do I get the answer in radians instead of degrees?
- Multiply the degree result by π ÷ 180. Since arcsin(0.5) = 30°, the same angle in radians is 30 × π ÷ 180 = π/6, roughly 0.5236. Calculators set to radian mode will report that value directly.