Inverse Sine CalculatorFree online math calculator — instant results, no sign-up
Calculate the principal inverse sine in degrees.
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About the Inverse Sine Calculator
The Inverse Sine Calculator converts a sine value into its principal angle in degrees. It accepts values from −1 to 1 and returns an angle from −90° to 90°. With the default sine value of 0.5, the result is 30°.
- Sine value
- The dimensionless sine ratio whose principal angle you want to find, with a default value of 0.5.
Formula and method
θ = arcsin(x) × 180/π; for x = 0.5, θ = (π/6) × 180/π = 30°.
- xThe input sine value, a dimensionless number between −1 and 1 inclusive.
- θThe principal inverse sine angle, expressed in degrees.
- arcsinThe inverse sine function, returning its principal angle in radians before conversion to degrees.
- πThe mathematical constant pi; π radians equals 180 degrees.
Check that the sine value lies within the real-valued domain [−1, 1]. Evaluate its principal inverse sine in radians, then multiply by 180/π to express the angle in degrees. For the default input, arcsin(0.5) = π/6 radians, so the displayed result is 30.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Sine value0.5
Results
- Result30
How to use the Inverse Sine Calculator
- 1Enter the sine ratio in the Sine value field, or leave the default value of 0.5.
- 2Confirm that the input is between −1 and 1 inclusive and is a ratio rather than an angle or percentage.
- 3Run the calculation to evaluate the principal inverse sine and convert it to degrees.
- 4Read the Result row: with the default input of 0.5, Result = 30, meaning 30°.
Reading your result
The Result row reports the principal angle in degrees whose sine equals the input. Result = 30 therefore means sin(30°) = 0.5. The returned angle is always within −90° to 90°; it is not a list of every angle with that sine value.
Limitations
Real-valued inverse sine is defined only for inputs from −1 to 1 inclusive. This calculator returns degrees and only the principal angle, not all periodic solutions or complex-valued results. Its accuracy depends on the precision of the supplied sine value, and any displayed rounding can limit the precision of the reported angle.
Common questions
Why does the default sine value of 0.5 give a result of 30?+
Because sin(30°) = 0.5, and 30° lies in the principal inverse sine range of −90° to 90°.
Does inverse sine mean the reciprocal of sine?+
No. Inverse sine, written arcsin(x) or sin⁻¹(x), finds an angle from a sine value. The reciprocal of sine is cosecant and is a different operation.
Why can I not enter a sine value outside −1 to 1?+
The sine of a real angle always lies between −1 and 1. An input outside that interval has no real inverse sine angle, so it cannot produce a valid real-valued result here.
Why does the calculator return only one angle when sine repeats?+
Inverse sine uses the restricted range −90° to 90° to select one unique principal angle. Finding every angle with the same sine requires additional symmetry and periodicity calculations.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.