Arcsin Calculator (Inverse Sine)

Enter a sine value (a ratio between -1 and 1) to find its inverse sine angle in degrees and radians, along with the matching cosine value and complementary angle.

Quick Facts

Domain and range
x ∈ [−1, 1] → θ ∈ [−90°, 90°]
arcsin is the inverse of sine restricted to this range, so it returns exactly one principal angle for each x.

Your Results

Calculated
Angle θ = arcsin(x)
-
Principal value, in degrees
Angle in radians
-
θ = arcsin(x), in radians
cos(θ) = √(1 − x²)
-
Cosine of the resulting angle
Complementary angle
-
90° − θ, equals arccos(x)

Ready

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

How the Arcsin Calculator (Inverse Sine) works

Arcsin, written arcsin(x) or sin⁻¹(x), is the inverse of the sine function. Sine takes an angle and returns a ratio between -1 and 1; arcsin takes that ratio and returns the angle that produced it. Since sine repeats every 360°, arcsin is defined to return only one answer per input — the principal value, between -90° and 90° (-π/2 and π/2 radians).

The formula

For any x with -1 ≤ x ≤ 1, this calculator computes:

  • θ = arcsin(x), the principal angle in degrees, converted from radians as θ° = θrad × 180/π.
  • θ in radians, computed directly using the JavaScript Math.asin(x) function.
  • cos(θ) = √(1 − x²), derived from the Pythagorean identity sin²θ + cos²θ = 1. Because arcsin's range is limited to -90°–90°, cosine there is never negative, so this square root is always valid.
  • The complementary angle, 90° − θ, which is exactly arccos(x) — a direct consequence of the identity arcsin(x) + arccos(x) = 90°.

Right-triangle interpretation

Arcsin has a direct geometric meaning. In a right triangle, sin(θ) = opposite ÷ hypotenuse. If you know that ratio, arcsin(x) tells you the angle θ itself. For example, a triangle with a hypotenuse of 5 and an opposite side of 3 has sin(θ) = 3/5 = 0.6, so θ = arcsin(0.6) ≈ 36.87°.

Domain and undefined inputs

Because sine never produces a value outside -1 to 1, arcsin is only defined for x in that same interval. Entering a value like 1.5 has no real solution and this calculator will flag it as invalid rather than return a result.

Degrees vs. radians

Radians are the native unit for trigonometric functions in mathematics and programming (one full circle is 2π radians), while degrees are more common in everyday and classroom use (a full circle is 360°). This calculator reports both, since either can be the one you need depending on context.

Frequently Asked Questions

What is the domain and range of the arcsin function?
The domain of arcsin(x) is -1 to 1, because sine values never fall outside that range. Its range (the principal value) is -90° to 90°, or -π/2 to π/2 radians, so arcsin always returns exactly one angle in that interval.
How is arcsin related to a right triangle?
In a right triangle, sin(θ) = opposite ÷ hypotenuse. Arcsin reverses this: given that ratio, arcsin returns the angle θ. For example, arcsin(0.5) = 30°, matching a 30-60-90 triangle where the side opposite the 30° angle is half the hypotenuse.
What is the difference between arcsin and arccos?
arcsin(x) and arccos(x) are complementary: arcsin(x) + arccos(x) = 90° (π/2 radians) for any x between -1 and 1. Once you know arcsin(x), you can find arccos(x) by subtracting it from 90°.
Why does arcsin only return values between -90° and 90°?
Sine is periodic, so many angles share the same sine value (sin(30°) and sin(150°) are both 0.5). To make arcsin a proper function with one output per input, it is restricted to the principal range of -90° to 90°; other solutions can be found by reflecting across 90° or adding multiples of 360°.

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