Arcsin Calculator (Inverse Sine)
Calculate the inverse sine (arcsin) of a value and get results in radians and degrees
Calculate Inverse Sine (arcsin)
Domain: -1 ≤ x ≤ 1
Range: -π/2 to π/2 (or -90° to 90°)
Arcsin Results
Function notation: arcsin(0) = sin⁻¹(0)
Calculation: arcsin(0) = 0.000000 radians = 0.00°
Special angle: This is a common trigonometric value = 0 radians
Related Functions
cos(arcsin(0)): 1.000000
tan(arcsin(0)): 0.000000
Input Domain
The arcsin function is only defined for values between -1 and 1 (inclusive).
- -1 ≤ x ≤ 1
- Example: -1, -0.5, 0, 0.5, 1
- -π/2 ≤ arcsin(x) ≤ π/2 radians
- -90° ≤ arcsin(x) ≤ 90°
Common Arcsin Values
| x | arcsin(x) (radians) | arcsin(x) (degrees) | Exact Value |
|---|---|---|---|
| -1 | -π/2 ≈ -1.571 | -90° | -π/2 |
| -√3/2 ≈ -0.866 | -π/3 ≈ -1.047 | -60° | -π/3 |
| -√2/2 ≈ -0.707 | -π/4 ≈ -0.785 | -45° | -π/4 |
| -1/2 = -0.5 | -π/6 ≈ -0.524 | -30° | -π/6 |
| 0 | 0 | 0° | 0 |
| 1/2 = 0.5 | π/6 ≈ 0.524 | 30° | π/6 |
| √2/2 ≈ 0.707 | π/4 ≈ 0.785 | 45° | π/4 |
| √3/2 ≈ 0.866 | π/3 ≈ 1.047 | 60° | π/3 |
| 1 | π/2 ≈ 1.571 | 90° | π/2 |
Quick Reference
Function
y = arcsin(x) or y = sin⁻¹(x)
Domain
-1 ≤ x ≤ 1
Range
-π/2 ≤ y ≤ π/2 (-90° to 90°)
Relationship
If y = arcsin(x), then sin(y) = x
Key Properties
arcsin is the inverse function of sine
Principal value range: [-π/2, π/2]
arcsin(-x) = -arcsin(x) (odd function)
arcsin(x) is an increasing function
arcsin(x) + arccos(x) = π/2
Applications
Finding angles in right triangles
Physics: projectile motion analysis
Wave analysis and signal processing
Optics: refraction angles
Forensics: angle of impact analysis
Understanding Arcsin (Inverse Sine)
What is Arcsin?
Arcsin (or inverse sine) is the inverse function of the sine function. It answers the question: "What angle has a sine value of x?" The notation arcsin(x) is equivalent to sin⁻¹(x).
Domain and Range
- •Domain: [-1, 1] - sine values are always between -1 and 1
- •Range: [-π/2, π/2] radians or [-90°, 90°] - principal value range
- •The function is strictly increasing within its domain
Mathematical Properties
sin(arcsin(x)) = x for x ∈ [-1, 1]
arcsin(sin(θ)) = θ for θ ∈ [-π/2, π/2]
arcsin(-x) = -arcsin(x)
Related Functions
cos(arcsin(x)) = √(1 - x²)
tan(arcsin(x)) = x / √(1 - x²)
arcsin(x) + arccos(x) = π/2
Common Applications
Right Triangle Analysis
If you know the opposite side and hypotenuse of a right triangle, arcsin helps find the angle: α = arcsin(opposite/hypotenuse).
Physics Applications
Used in optics for refraction calculations, projectile motion analysis, and wave physics to determine angles from amplitude ratios.
Step-by-Step Example
Problem: Find arcsin(0.5)
Step 1: Verify input is in domain [-1, 1] ✓
Step 2: Apply arcsin function
Step 3: arcsin(0.5) = π/6 radians = 30°
Verification: sin(30°) = sin(π/6) = 0.5 ✓
Triangle context: In a 30-60-90 triangle, the angle opposite the side of length 1 (when hypotenuse = 2) is 30°