Last updated: July 31, 2026
Inverse Sine Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
arcsin(x) = sin^-1(x) = Math.asin(x), and degrees = radians × 180/π for -1 ≤ x ≤ 1.
Where:
- x=Input value(unitless)
- θ=Principal inverse sine angle(radians or degrees)
- π=Pi constant(unitless)
Worked Examples
Half on the unit interval
x = 0.5 corresponds to a standard special angle.
- 1Check that 0.5 lies within the valid domain.
- 2Compute arcsin(0.5) = π/6.
- 3Convert π/6 radians to 30 degrees.
Maximum valid value
The top of the domain returns the highest principal angle.
- 1Confirm x = 1 is allowed.
- 2Compute arcsin(1) = π/2.
- 3Convert π/2 to degrees.
Zero input
arcsin(0) is the neutral angle at the origin.
- 1Use x = 0 within the valid domain.
- 2Compute arcsin(0) = 0.
- 3The degree value remains 0.
Domain error example
Values outside [-1, 1] are not valid inputs for arcsin.
- 1Notice that 1.5 is greater than 1.
- 2The inverse sine function is undefined there for real numbers.
- 3The calculator returns an error instead of a complex result.
Introduction
The inverse sine calculator converts a valid sine value back into its principal angle in both radians and degrees. It accepts real inputs only on the interval from -1 to 1, which is the real domain of arcsin. This makes it useful for trigonometry practice, unit-circle problems, and checking calculator work.
What inverse sine means
Inverse sine answers the question: which principal angle has sine equal to x?
It reverses the sine function on a restricted interval.
The input is a sine value, not an angle.
The output is the principal angle.
Results can be shown in radians and degrees.
It is commonly written as arcsin(x) or sin⁻¹(x).
The calculator uses JavaScript’s real-valued asin function.
Domain and range
For real-valued output, the input must lie between -1 and 1 inclusive.
Valid inputs satisfy -1 ≤ x ≤ 1.
x = -1 maps to -90°.
x = 0 maps to 0°.
x = 1 maps to 90°.
The principal range is -90° to 90°.
Values outside the domain are undefined over the reals.
The calculator does not return complex-number results for out-of-domain inputs.
Why degrees are shown
Many classroom and applied problems are stated in degrees, even though the core inverse function returns radians.
The calculator first computes radians.
It multiplies radians by 180/π.
Degrees are easier to interpret visually.
Radians remain useful for calculus and analysis.
Both outputs support cross-checking.
Rounded values stay readable.
How to use the calculator
Only one numeric input is needed for inverse sine.
Enter a value for x.
Keep x inside [-1, 1].
Submit the calculation.
Read radians for exact trig workflows.
Read degrees for everyday angle interpretation.
Watch the error message if the domain fails.
Interpreting the result
The radian and degree outputs describe the same principal angle in two units.
Radians are the native output of arcsin.
Degrees are a direct conversion.
A positive result means a positive principal angle.
A negative result means a negative principal angle.
Zero means the horizontal baseline.
Special values often match known unit-circle angles.
Common mistakes
Users often confuse the input and output roles of sine and inverse sine.
Entering an angle instead of a sine value.
Forgetting that x must stay within [-1, 1].
Mixing radians and degrees.
Expecting more than one principal answer.
Rounding before converting units.
Reading sin⁻¹(x) as 1/sin(x).
Where inverse sine is used
Inverse sine appears in geometry, trigonometry, physics, and data analysis whenever a ratio must be converted into an angle.
Solving right-triangle problems.
Recovering angles from normalized measurements.
Studying unit-circle relationships.
Checking graphing-calculator output.
Building trigonometric code.
Verifying classroom examples.
FAQs
What is the domain of arcsin(x)?
For real outputs, x must be between -1 and 1 inclusive.
What is the range of the inverse sine output?
The principal range is from -90° to 90°, or from -π/2 to π/2 radians.
Why do I get an error for x = 1.5?
Because 1.5 is outside the real-valued domain of inverse sine.
Does the calculator show radians and degrees?
Yes. It returns the principal angle in both radians and degrees.
What does sin⁻¹(x) mean?
It means inverse sine, not one divided by sine.
Can the output be negative?
Yes. Negative input values can produce negative principal angles.
Why is 0.5 a special example?
Because arcsin(0.5) equals 30°, a standard unit-circle value.