Understanding the Arcsin Function (Inverse Sine)
The arcsin function, written as arcsin(x) or sin-1(x), is the inverse of the sine function. Given a sine value, it returns the angle whose sine is that value.
Domain and Range
Domain of arcsin: [-1, 1]
Range of arcsin: [-π/2, π/2] radians = [-90°, 90°]
If θ = arcsin(x), then sin(θ) = x and -π/2 ≤ θ ≤ π/2
Range of arcsin: [-π/2, π/2] radians = [-90°, 90°]
If θ = arcsin(x), then sin(θ) = x and -π/2 ≤ θ ≤ π/2
This restricted range is the principal value. On [-π/2, π/2], sine is strictly increasing, making the inverse unique.
Common Arcsin Values
| x | Degrees | Radians | Exact |
|---|---|---|---|
| -1 | -90° | -1.57080... | -π/2 |
| -√3/2 | -60° | -1.04720... | -π/3 |
| -√2/2 | -45° | -0.78540... | -π/4 |
| -1/2 | -30° | -0.52360... | -π/6 |
| 0 | 0° | 0 | 0 |
| 1/2 | 30° | 0.52360... | π/6 |
| √2/2 | 45° | 0.78540... | π/4 |
| √3/2 | 60° | 1.04720... | π/3 |
| 1 | 90° | 1.57080... | π/2 |
Relationship with Arccos
arcsin(x) + arccos(x) = π/2
arcsin(x) = π/2 - arccos(x)
arcsin(-x) = -arcsin(x) (odd function)
arcsin(x) = π/2 - arccos(x)
arcsin(-x) = -arcsin(x) (odd function)
Derivative and Integral
d/dx [arcsin(x)] = 1 / sqrt(1 - x²) for |x| < 1
∫ arcsin(x) dx = x · arcsin(x) + sqrt(1 - x²) + C
∫ arcsin(x) dx = x · arcsin(x) + sqrt(1 - x²) + C
Applications
- Physics: Launch angles in projectile motion. Snell's law of refraction.
- Navigation: The haversine formula for great-circle distances involves arcsin.
- Statistics: The arcsine transformation arcsin(√p) stabilizes variance of proportions.
- Geometry: Finding angles using the law of sines: A = arcsin(a · sin(B) / b).
- Engineering: Signal processing phase calculations.