Inverse Sine Calculator: Calculate sin⁻¹ in Degrees or Radians

An inverse sine calculator finds the angle whose sine is a number you enter. Enter a value from −1 to 1 to get the principal angle in degrees or radians, along with its equivalent in the other unit.

Formula

The inverse sine, written sin⁻¹(x) or arcsin(x), returns the principal angle whose sine is x. For an answer in degrees, use θ = arcsin(x) × 180/π. For an answer in radians, use θ = arcsin(x). Here, x is the sine value entered, with no physical unit, and θ is an angle. The calculator’s unit choice selects the headline result; the other two outputs show the corresponding angle in degrees and radians.

The valid input range is −1 ≤ x ≤ 1 because the sine of a real angle cannot be outside that interval. The principal-value range is −90° to 90°, or −π/2 to π/2 radians. This consistent range is important: inverse sine returns one chosen angle, not every angle that has the given sine.

Worked Example

Using the defaults, the input is x = 0.5 and the selected unit is degrees. Since sin(30°) = 0.5, the principal inverse sine is 30°.

• Degree result: arcsin(0.5) × 180/π = 30°.
• Radians result: arcsin(0.5) = π/6 ≈ 0.5236 radians.

Rounded as shown by the calculator, the headline answer is 30.00 degrees, with the equivalent angle 30.00 degrees or 0.5236 radians.

How to Calculate Inverse Sine

For a value that is not one of the familiar sine values, use a calculator with an arcsin or sin⁻¹ function. The steps are:

1. Check that the input is between −1 and 1, inclusive.
2. Choose whether the result should be in degrees or radians.
3. Enter the sine value and evaluate arcsin, asin, or sin⁻¹.
4. Keep enough digits for later calculations, and round only when presenting the result.

For example, arcsin(0.3) is approximately 17.46° or 0.3047 radians. The sine value itself has no angle unit; the output does. A calculator’s angle-mode setting can affect other trigonometric operations, so check that setting when using a handheld calculator. The inverse-sine function may have its own output convention or follow the device’s degree/radian mode.

What the Result Means

The result answers the question, “Which principal angle has this sine?” For x = 0, the answer is 0°. For x = 1, it is 90° or π/2 radians; for x = −1, it is −90° or −π/2 radians. Positive inputs give nonnegative principal angles, and negative inputs give nonpositive principal angles.

Inverse sine is useful when a right-triangle calculation gives the ratio of the opposite side to the hypotenuse. If that ratio is 0.5, the corresponding acute angle is 30°. In a broader geometry or physics problem, arcsin can also recover an angle from a sine-based relationship, provided the input is in range and the intended solution is the principal one.

Multiple Angles and Common Mistakes

Sine repeats, so many angles can share the same sine. For example, 30° and 150° both have sine 0.5. The inverse-sine function returns 30°, the principal value; it does not list 150° or angles differing by full turns. In a general equation sin(θ) = x, solutions repeat every 360° (2π radians), with a second family reflected across 90° (π/2 radians). Apply the problem’s angle range and other conditions to select the relevant solution.

• Using an invalid input: A real-valued arcsin is undefined for inputs below −1 or above 1. Small out-of-range results caused by rounding may require checking the source data rather than blindly changing the value.
• Confusing inverse sine with a reciprocal: sin⁻¹(x) means arcsin(x), not 1/sin(x). The reciprocal is cosecant, written csc(x).
• Mixing degrees and radians: 1 radian is 180/π degrees, approximately 57.2958°. A value such as 0.5236 is about 30°, not 0.5236°.
• Rounding too early: Keep extra digits during multi-step work. Rounding an intermediate angle can shift a later result.

Limits and Related Functions

Inverse sine is one-to-one only after its output is restricted to the principal range. That restriction makes it a well-defined inverse of sine on −90° to 90°. It does not mean that sine is one-to-one over all real angles; periodicity is why a full equation can have multiple solutions.

Use inverse cosine, arccos(x), when the known ratio is adjacent side divided by hypotenuse, and inverse tangent, arctan(x), when it is opposite side divided by adjacent side. For two known coordinates or components, atan2 is often more suitable than arctan of a ratio because it can distinguish the angle’s quadrant; its argument order and units depend on the function or tool being used.

Frequently Asked Questions

What is the inverse sine of 0.5?
Its principal value is 30°, which is π/6 radians (about 0.5236 radians).

How do I calculate sin⁻¹ on a calculator?
Use the arcsin or sin⁻¹ function, enter a value from −1 to 1, and check whether the result is set to degrees or radians.

Is sin⁻¹ the same as 1/sin?
No. sin⁻¹(x) usually means arcsin(x), the angle with sine x. The reciprocal of sin(x) is csc(x).

Why does inverse sine give only one angle?
Arcsin is defined to return a principal angle between −90° and 90°. Other angles can have the same sine, so solve the full equation separately if all solutions are needed.

Can I take the inverse sine of 2?
Not as a real-valued angle. A real sine value cannot exceed 1 or be less than −1, so arcsin(2) has no real result.

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in x | Sine value | - | 0.5 | -1
sel u | Angle unit | Degrees=1, Radians=2
out angle | Inverse sine | selected unit | asind(x) | 2
out degrees | Angle | degrees | asind(x) | 2
out radians | Angle | radians | asin(x) | 4