Inverse Sine Calculator - Find arcsin(x) = sin⁻¹(x)

Using this sin⁻¹ calculator is straightforward:

1. Enter your value — type any number between -1 and 1, or a fraction like 45/89, 7/12, or 4/6. The calculator accepts both decimal and fractional input.
2. Choose degrees or radians — select your preferred output unit. Both are always shown in the results.
3. Set precision — choose 0 to 8 decimal places depending on how precise you need the answer.
4. Click Calculate or press Enter — the result appears instantly with the angle in degrees, radians, and DMS (degrees-minutes-seconds) format.

The calculator also shows exact values for standard angles (like arcsin(0.5) = 30° = π/6), related trig values cos(θ) and tan(θ), and the general solution formula for all angles with the same sine. For the forward calculation — when you know the angle and need the sine value — use our sin calculator.

The inverse sine function reverses the sine function. While sin(θ) takes an angle and returns a ratio, arcsin(x) takes a ratio and returns the angle. The mathematical definition is:

The notation sin⁻¹(x) means "the angle whose sine is x" — the superscript -1 is not an exponent. It is the same as arcsin(x), just a different way to write it.

Domain and Range of arcsin

• Domain: x ∈ [-1, 1] — because sine values only range from -1 to 1, you cannot compute arcsin of a number outside this interval.
• Range (principal value): θ ∈ [-π/2, π/2] = [-90°, 90°] — arcsin always returns the angle in this range to ensure a unique answer.

For example, sin(30°) = 0.5 and sin(150°) = 0.5 — both give the same sine value. The arcsin function returns only the principal value of 30° (π/6). To find all solutions, use the general solution: θ = nπ + (-1)ⁿ · arcsin(x), where n is any integer.

These are the most frequently searched sin inverse values from the unit circle. Memorizing these helps with quick mental calculations:

Notice the symmetry: arcsin(-x) = -arcsin(x). So if you know arcsin(0.5) = 30°, then arcsin(-0.5) = -30°. The values 0.707 (√2/2) and 0.866 (√3/2) appear constantly in physics and engineering problems involving 45° and 60° angles.

Here is how to find sin⁻¹(x) manually and verify your results:

Example: Find arcsin(0.5) in Degrees and Radians

1. Check the domain: Is 0.5 between -1 and 1? Yes — proceed.
2. Recall or calculate: We need the angle θ where sin(θ) = 0.5. From the unit circle, sin(30°) = 0.5.
3. Result in degrees: arcsin(0.5) = 30°
4. Convert to radians: 30° × (π/180°) = π/6 ≈ 0.5236 rad
5. Verify: sin(30°) = sin(π/6) = 0.5 ✓

Example: Find sin⁻¹(7/12) — Fraction Input

1. Evaluate the fraction: 7/12 ≈ 0.5833
2. Check domain: 0.5833 is between -1 and 1 ✓
3. Calculate: arcsin(0.5833) ≈ 35.685° ≈ 0.6228 rad
4. In DMS: 35° 41′ 6.00″

This is exactly the kind of calculation students and professionals do daily — our sin inverse calculator handles these instantly, including fraction inputs like sin⁻¹(45/89), sin⁻¹(7/9.3), and sin⁻¹(8/12) that appear frequently in right-triangle problems.

One of the most common questions is whether to use degrees or radians for inverse sine results. The answer depends on your context:

• Degrees — use in construction, surveying, navigation, and most everyday applications. A full circle = 360°.
• Radians — use in calculus, physics formulas, programming, and higher mathematics. A full circle = 2π rad.

Our calculator outputs both simultaneously, so you never need to convert manually. For dedicated angle conversion, see our degrees to radians calculator or radians to degrees calculator.

The DMS format (degrees, minutes, seconds) is also provided — this is the standard notation in surveying, astronomy, and geographic coordinates. For example, arcsin(0.707) = 44° 59′ 51.13″ ≈ 45°.

Construction and Architecture

When a roof has a rise-to-run ratio, the inverse sine converts this ratio to the pitch angle needed for building permits and material calculations. A rise/hypotenuse ratio of 0.5 yields arcsin(0.5) = 30° — a standard roof pitch.

Physics: Projectile Motion and Optics

In projectile motion, the launch angle θ satisfies sin(2θ) = gR/v², requiring arcsin to solve for θ. In optics, Snell's law (n₁·sin(θ₁) = n₂·sin(θ₂)) uses arcsin to find the refraction angle: θ₂ = arcsin((n₁/n₂)·sin(θ₁)). The inverse sine is also critical for finding phase angles in AC circuit analysis.

Right Triangle Problems (SOHCAHTOA)

Given opposite = 7 and hypotenuse = 12 in a right triangle, the angle is arcsin(7/12) ≈ 35.69°. This is why so many people search for values like sin⁻¹(45/89), sin⁻¹(7/12), or sin⁻¹(4/6) — they're solving triangle problems. For full triangle analysis, our right triangle calculator and SOHCAHTOA calculator provide comprehensive solutions.

Navigation and Surveying

Navigators use arcsin to convert elevation ratios to angles of inclination. A trail with a 0.707 elevation ratio represents a 45° incline — critical information for route planning. For full trigonometric analysis, our trigonometry calculator handles all six trig functions and their inverses.

These are all the same function with different notation used in different contexts:

• arcsin(x) — standard mathematical notation, most common in textbooks and academic writing.
• sin⁻¹(x) — scientific calculator notation. The -1 is not an exponent — it means "inverse function," not 1/sin(x).
• asin(x) — programming notation used in JavaScript, Python, C/C++, and most programming languages (e.g., Math.asin()).

A common mistake: sin⁻¹(x) ≠ 1/sin(x). The reciprocal of sine is cosecant (csc(x) = 1/sin(x)), not the inverse sine. This distinction trips up many students.

For the related inverse functions, see our inverse cosine calculator (arccos) and inverse tangent calculator (arctan). You can also use our arcsin calculator for an alternative interface focused on the arcsin notation.

The arcsin function returns only the principal value — the unique angle in [-90°, 90°]. But sine is periodic, so infinitely many angles share the same sine value. The general solution captures all of them:

For example, if arcsin(0.5) = 30°, the full set of solutions is: ..., -330°, -210°, 30°, 150°, 390°, 510°, ... This matters in physics (wave equations), engineering (signal processing), and any context where periodicity plays a role.

• Fraction support — enter 7/12, 45/89, or any fraction directly without converting to decimal first.
• Dual output — every result shows degrees, radians, and DMS simultaneously.
• Exact value recognition — automatically identifies standard angles (30°, 45°, 60°) and their exact radian values (π/6, π/4, π/3).
• Built-in verification — confirms sin(result) = your input, so you can trust the answer.
• General solution — shows the formula for all angles with the same sine value, not just the principal value.
• Related trig values — displays cos(θ) and tan(θ) at the computed angle for complete analysis.
• Mobile-friendly — fully responsive design with touch-friendly buttons (minimum 44px tap targets).

Bookmark this page for quick access whenever you need to compute sin⁻¹(x), arcsin(x), or any inverse sine calculation in degrees or radians.