Inverse Cosine (arccos) Calculator — Degrees, Radians & More
Calculate the inverse cosine (arccos) of any value. Get results in degrees, radians, gradians, and turns with domain checking, related trig values, and a common values reference table.
Inverse Sine (arcsin) Calculator — Degrees, Radians & Unit Circle
Calculate the inverse sine (arcsin) of any value. Get results in degrees, radians, gradians, and turns with domain checking, unit circle visualization, and all 6 inverse trig comparisons.
Must be between −1 and 1
Degrees⧉
30.000000°
arcsin(x) in degrees, range [−90°, 90°]
Radians⧉
0.523599
arcsin(x) in radians, range [−π/2, π/2]
Gradians⧉
33.333333
arcsin(x) in gradians (400 grad = 360°)
Turns⧉
0.083333
Fraction of a full revolution (1 turn = 360°)
Complement (arccos)⧉
60.000000°
arccos(x) = 90° − arcsin(x)
Verification sin(θ)⧉
0.500000
sin(arcsin(x)) should equal x — round-trip check
arctan(x)⧉
26.565051°
Inverse tangent of the same input
Domain Status⧉
✓ Valid
arcsin is defined only for x ∈ [−1, 1]
Result Visualization
Input x
0.500
arcsin(x)
30.0°
Unit Circle Position
All Inverse Trig Functions at x = 0.5000
| Function | Result (°) | Domain | Range | Status |
|---|---|---|---|---|
| arcsin(x) | 30.000000° | [−1, 1] | [−90°, 90°] | ✓ Valid |
| arccos(x) | 60.000000° | [−1, 1] | [0°, 180°] | ✓ Valid |
| arctan(x) | 26.565051° | (−∞, ∞) | (−90°, 90°) | ✓ Valid |
| arccsc(x) | — | |x| ≥ 1 | [−90°, 90°]\{0} | ✗ Outside |
| arcsec(x) | — | |x| ≥ 1 | [0°, 180°]\{90°} | ✗ Outside |
| arccot(x) | 63.434949° | x ≠ 0 | (0°, 180°) | ✓ Valid |
Common arcsin Values
| x | Degrees | Exact Radians | Decimal Radians |
|---|---|---|---|
| −1 | −90° | −π/2 | −1.5708 |
| −√3/2 ≈ −0.8660 | −60° | −π/3 | −1.0472 |
| −√2/2 ≈ −0.7071 | −45° | −π/4 | −0.7854 |
| −1/2 = −0.5 | −30° | −π/6 | −0.5236 |
| 0 | 0° | 0 | 0 |
| 1/2 = 0.5 | 30° | π/6 | 0.5236 |
| √2/2 ≈ 0.7071 | 45° | π/4 | 0.7854 |
| √3/2 ≈ 0.8660 | 60° | π/3 | 1.0472 |
| 1 | 90° | π/2 | 1.5708 |
Domain & Range Reference
| Property | Value |
|---|---|
| Function | y = arcsin(x) = sin⁻¹(x) |
| Domain | [−1, 1] |
| Range | [−π/2, π/2] or [−90°, 90°] |
| Identity | arcsin(x) + arccos(x) = π/2 |
| Odd function | arcsin(−x) = −arcsin(x) |
| Derivative | d/dx arcsin(x) = 1/√(1 − x²) |
| At x = 0 | arcsin(0) = 0 |
| At x = 1 | arcsin(1) = π/2 = 90° |
| At x = −1 | arcsin(−1) = −π/2 = −90° |
Planning notes, formulas, and examples
About the Inverse Sine (arcsin) Calculator — Degrees, Radians & Unit Circle
The **Inverse Sine (arcsin) Calculator** finds the angle whose sine equals a given value. Enter any number between −1 and 1 — as a decimal or fraction — and obtain the result in degrees, radians, gradians, and turns, along with a unit circle visualization that shows exactly where the angle sits.
The arcsin function is one of the six fundamental inverse trigonometric functions. It reverses the sine operation: given a ratio of opposite side to hypotenuse, arcsin tells you the corresponding angle. This makes it essential in right-triangle problems, physics (projectile launch angles, wave phase), engineering (signal processing, structural analysis), and navigation (altitude angles).
Beyond the primary result, this calculator evaluates all six inverse trig functions — arcsin, arccos, arctan, arccsc, arcsec, and arccot — at the same input, so you can compare their values and see at a glance which functions accept the current input and which do not. A complementary identity display confirms that arcsin(x) + arccos(x) = 90° for every valid input.
Eight preset buttons load the standard unit-circle values from −90° to 90°. A fraction input mode lets you type exact values like √3/2 without rounding. Visual progress bars map the input across its domain and the output across its range. Reference tables list all common arcsin values and domain/range properties for quick lookup.
When This Page Helps
Inverse Sine (arcsin) Calculator — Degrees, Radians & Unit Circle helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Degrees, Radians, Gradians in one pass.
How to Use the Inputs
- Enter the required inputs (Input Mode, Value (x), Numerator).
- Complete the remaining fields such as Denominator, Decimal Precision.
- Review the output cards, especially Degrees, Radians, Gradians, Turns.
- Compare the result with the formula, diagram, or example values to catch sign, unit, or rounding mistakes.
Formula used
θ = arcsin(x), where x ∈ [−1, 1] and θ ∈ [−π/2, π/2]. In degrees: θ° = arcsin(x) × 180/π. Identity: arcsin(x) + arccos(x) = π/2. Odd symmetry: arcsin(−x) = −arcsin(x). Derivative: d/dx arcsin(x) = 1/√(1 − x²).Example Calculation
Result: 30°
Using value=0.5, unit=degrees, the calculator returns 30°. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
Tips & Best Practices
- The domain of arcsin is [−1, 1] — any input outside this range is out of bounds.
- arcsin is an odd function: arcsin(−x) = −arcsin(x), so negative inputs give negative angles.
- Use the identity arcsin(x) + arccos(x) = 90° to derive arccos directly from arcsin.
- In JavaScript, Math.asin(x) returns radians. Multiply by 180/Math.PI for degrees.
- For projectile problems, the launch angle θ satisfies sin θ = h/v₀t — use arcsin to solve.
What This Inverse Sine (arcsin) Calculator — Degrees, Radians & Unit Circle Solves
This calculator is tailored to inverse sine (arcsin) calculator — degrees, radians & unit circle workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
How To Interpret The Outputs
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
Study And Practice Strategy
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
Sources & Methodology
Last updated:
Frequently Asked Questions
The domain is [−1, 1]. Since sine outputs values only in this interval, only these values can be reversed. Any input outside [−1, 1] has no real arcsin value.
The principal-value range is [−π/2, π/2] or [−90°, 90°]. This is the branch chosen so that arcsin is a one-to-one function.
They are complementary: arcsin(x) + arccos(x) = π/2 (90°) for all x in [−1, 1]. If one is known, subtract from 90° to get the other.
No. arcsin(x) = sin⁻¹(x) is the inverse function, while 1/sin(x) = csc(x) is the reciprocal. The superscript −1 denotes the inverse, not an exponent.
Because the range includes negative values: [−90°, 90°]. Negative inputs yield negative angles by the odd-function property arcsin(−x) = −arcsin(x).
The general solution is θ = arcsin(x) + 360°n or θ = 180° − arcsin(x) + 360°n for any integer n. Arcsin returns only the principal value in [−90°, 90°].
More in this topic
Inverse Tangent (arctan) Calculator — arctan & atan2 Modes
Calculate arctan(x) or atan2(y,x) with full quadrant support. Get results in degrees, radians, gradians, and turns with quadrant identification, common values table, and atan-vs-atan2 comparison.
Inverse Trigonometric Functions Calculator — All 6 at Once
Calculate all six inverse trig functions simultaneously: arcsin, arccos, arctan, arccsc, arcsec, arccot. Enter a value and see every result with domain checking, identities, and visual comparison.