What Is Arcsine?
Arcsine (written arcsin(x) or sin⁻¹(x)) is the inverse of the sine function. Given a known sine value x, it returns the angle θ whose sine equals x:
arcsin(x) = θ iff sin(θ) = x
Because sine is periodic, infinitely many angles share the same sine value. For example, sin(30°) = sin(150°) = 0.5. To give a unique answer, arcsine returns only the principal value — the angle in the restricted range −90° ≤ θ ≤ 90° (or −π/2 ≤ θ ≤ π/2 in radians).
The domain of arcsine is −1 ≤ x ≤ 1 (real-number inputs only). Values outside this range have no real arcsine.
Graph of arcsin(x)
The graph of y = arcsin(x) is an S-shaped curve passing through the origin. Key points on the graph include:
- arcsin(−1) = −90° (= −π/2 rad)
- arcsin(−√3/2) = −60° (= −π/3 rad)
- arcsin(−√2/2) = −45° (= −π/4 rad)
- arcsin(−1/2) = −30° (= −π/6 rad)
- arcsin(0) = 0°
- arcsin(1/2) = 30° (= π/6 rad)
- arcsin(√2/2) = 45° (= π/4 rad)
- arcsin(√3/2) = 60° (= π/3 rad)
- arcsin(1) = 90° (= π/2 rad)
Inverse Sine, Trigonometric Functions, and Other Relationships
Arcsine is related to other inverse trigonometric functions through several identities:
- arcsin(x) + arccos(x) = π/2 (complementary relationship)
- arcsin(−x) = −arcsin(x) (odd function)
- arcsin(x) = arctan(x / √(1 − x²)) for |x| < 1
- d/dx arcsin(x) = 1 / √(1 − x²) (derivative)
- ∫ arcsin(x) dx = x·arcsin(x) + √(1 − x²) + C (integral)
Note: The notation sin⁻¹(x) means the inverse function, not 1/sin(x).
The abbreviation asin(x) is used in most programming languages.
Example: How to Use the Arcsin Calculator
- Enter the sine value in the input field (any number from −1 to 1). For example, type 0.5.
- Click Calculate.
- The calculator instantly shows the angle in degrees, radians, radians as a multiple of π, and gradians.
- For x = 0.5: arcsin(0.5) = 30° = π/6 rad ≈ 0.5236 rad = 33.333 grad.
- The Verification line confirms that sin(30°) = 0.5 ✓
Use the Common Values quick-buttons to jump straight to standard angles like 0, ±30°, ±45°, ±60°, ±90°.
Common Arcsin Values Table
| x (sine value) | arcsin(x) — Degrees | arcsin(x) — Radians | arcsin(x) — Gradians |
|---|---|---|---|
| −1 | −90° | −π/2 | −100 grad |
| −√3/2 ≈ −0.8660 | −60° | −π/3 | −66.667 grad |
| −√2/2 ≈ −0.7071 | −45° | −π/4 | −50 grad |
| −1/2 = −0.5 | −30° | −π/6 | −33.333 grad |
| 0 | 0° | 0 | 0 grad |
| 1/2 = 0.5 | 30° | π/6 | 33.333 grad |
| √2/2 ≈ 0.7071 | 45° | π/4 | 50 grad |
| √3/2 ≈ 0.8660 | 60° | π/3 | 66.667 grad |
| 1 | 90° | π/2 | 100 grad |
FAQs
What is the difference between arcsin and sin⁻¹?
They are the same thing. arcsin(x) and sin⁻¹(x) both denote the inverse sine function. The sin⁻¹ notation can be confusing because it may look like a reciprocal (1/sin x), so mathematicians often prefer "arcsin".
Why is the domain of arcsin limited to [−1, 1]?
The sine of any real angle always lies between −1 and 1. There is no real angle whose sine is, say, 2 — so arcsin(2) is undefined in the real numbers.
Can arcsin return an angle greater than 90°?
The principal value of arcsin is always in [−90°, 90°]. Other angles with the same sine exist (e.g., sin(150°) = sin(30°) = 0.5), but they are not the principal value returned by this calculator.
How are degrees, radians, and gradians related?
360° = 2π rad = 400 grad. Degrees split a full circle into 360 parts; radians measure arc length on a unit circle; gradians divide a right angle into 100 equal parts (used in surveying). This calculator shows all three for easy conversion.
What is the derivative of arcsin(x)?
d/dx arcsin(x) = 1 / √(1 − x²), defined for |x| < 1. This formula is widely used in integration and differential equations involving trigonometric substitutions.