Sin-1 Calculator

Enter a value x between -1 and 1 to find sin⁻¹(x) — the inverse sine (arcsine) — in degrees and radians, on the principal branch.

Quick Facts

Domain & Range
-1 ≤ x ≤ 1 → -90° to 90°
sin⁻¹(x) is only defined for x in [-1, 1] and returns the principal value in [-90°, 90°].
Key Identity
sin(sin⁻¹(x)) = x
Arcsine undoes sine on its restricted, one-to-one branch.
Derivative
d/dx[sin⁻¹(x)] = 1/√(1-x²)
Used in calculus for differentiation, integration, and related-rate problems.

Your Results

Calculated
Angle (Degrees)
-
θ = sin⁻¹(x), principal value
Angle (Radians)
-
θ in radians, -π/2 to π/2
Complementary Angle
-
90° - θ = cos⁻¹(x)
Verification: sin(θ)
-
Should equal your input x

Ready

Enter a value between -1 and 1, then press Calculate.

Formula and Method for the Inverse Sine (sin⁻¹) Calculator

The inverse sine function, written sin⁻¹(x) or arcsin(x), answers the question "which angle θ has a sine equal to x?" Because sin(θ) repeats forever, the inverse can only be a true function if its output is restricted to one interval — the principal branch. By convention, sin⁻¹(x) always returns an angle θ such that -90° ≤ θ ≤ 90° (equivalently -π/2 ≤ θ ≤ π/2 radians), and the input x must satisfy -1 ≤ x ≤ 1, since sine itself never exceeds that range. This calculator applies that standard convention (the same one used by scientific calculators, Excel's ASIN, and JavaScript's Math.asin).

How the calculation works

Enter a value x between -1 and 1. The calculator computes θ = sin⁻¹(x) in radians internally, then converts to degrees with θ° = θ_rad × 180/π. It also reports the complementary angle 90° - θ, which equals cos⁻¹(x) because sine and cosine of complementary angles are equal, and a verification value sin(θ) that should recompute back to your original x — a quick self-check that the result is correct.

Common mistakes

  • Entering a value outside [-1, 1]: sin⁻¹(2), for example, is undefined for real numbers because no angle has a sine of 2 — this calculator flags out-of-domain inputs.
  • Confusing sin⁻¹(x) with 1/sin(x): the "-1" here denotes function inversion (arcsine), not a reciprocal exponent. The reciprocal of sine is cosecant, csc(x) = 1/sin(x).
  • Forgetting the principal-value restriction: infinitely many angles share the same sine (e.g., sin(30°) = sin(150°) = 0.5), but sin⁻¹(0.5) always returns only 30°, the principal value.
  • Mixing degrees and radians: if you paste this result into another formula, make sure that formula expects the same unit (degrees or radians) that you read off here.

Real-world applications

  • Solving right triangles: if you know the opposite side and hypotenuse, θ = sin⁻¹(opposite/hypotenuse) gives the angle directly.
  • Physics and engineering: resolving projectile launch angles, incline angles, and wave phase shifts from a known ratio.
  • Navigation and surveying: back-calculating bearing or elevation angles from measured ratios of distances.
  • Signal processing and optics: computing angles of incidence or phase angles from amplitude ratios (e.g., Snell's law problems).

Frequently Asked Questions

What does sin⁻¹(x), or arcsin, actually mean?
sin⁻¹(x), also written arcsin(x), is the inverse sine function: it answers the question "which angle θ has sin(θ) = x?" It returns the principal value θ in the range -90° to 90° (-π/2 to π/2 radians).
What is the domain and range of sin⁻¹(x)?
The domain is -1 ≤ x ≤ 1, because sine never produces a value outside that interval. The range (the output) is restricted to -90° ≤ θ ≤ 90°, or -π/2 ≤ θ ≤ π/2 radians, which is the principal branch used by calculators and programming languages.
Is sin⁻¹(x) the same as 1/sin(x)?
No. sin⁻¹(x) is the inverse function (arcsine), while 1/sin(x) is the reciprocal, called cosecant, csc(x). The "-1" superscript on an inverse trig function denotes function inversion, not an exponent.
How do I convert the sin⁻¹ result between degrees and radians?
Multiply radians by 180/π to get degrees, or multiply degrees by π/180 to get radians. Most programming languages' Math.asin() returns radians, so multiply by 180/π before displaying degrees.

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