Sin⁻¹ Calculator (Arcsine)
Calculate the inverse sine (arcsine) function and convert between radians and degrees
Calculate Inverse Sine
Domain: -1 ≤ x ≤ 1 (sine values must be between -1 and 1)
Arcsine Results
Exact Value Recognized
sin⁻¹(Zero) = 0° = 0 rad
Formula Used
Input: x = 0
Formula: sin⁻¹(x) = y, where sin(y) = x
Domain: -1 ≤ x ≤ 1
Range: -π/2 ≤ y ≤ π/2 (-90° ≤ y ≤ 90°)
Result: y = 0.000000 rad = 0.000000°
Step-by-Step Solution
- Step 1: Verify that input x = 0 is in domain [-1, 1] ✓
- Step 2: Apply arcsine function: sin⁻¹(0)
- Step 3: Calculate result in radians: 0.000000 rad
- Step 4: Convert to degrees: 0.000000 × (180/π) = 0.000000°
- Step 5: Verify: sin(0.000000°) ≈ 0.000000 ≈ 0
Common Arcsine Values
| x | Description | sin⁻¹(x) (degrees) | sin⁻¹(x) (radians) |
|---|---|---|---|
| -1.000 | Minimum value | -90° | -π/2 |
| -0.866 | -√3/2 | -60° | -π/3 |
| -0.707 | -√2/2 | -45° | -π/4 |
| -0.500 | -1/2 | -30° | -π/6 |
| 0.000 | Zero | 0° | 0 |
| 0.500 | 1/2 | 30° | π/6 |
| 0.707 | √2/2 | 45° | π/4 |
| 0.866 | √3/2 | 60° | π/3 |
| 1.000 | Maximum value | 90° | π/2 |
Arcsine Properties
Domain
[-1, 1]
Range
[-π/2, π/2] or [-90°, 90°]
Symmetry
Odd function: sin⁻¹(-x) = -sin⁻¹(x)
Key Points
sin⁻¹(0) = 0°
sin⁻¹(1) = 90° = π/2 rad
sin⁻¹(-1) = -90° = -π/2 rad
sin⁻¹(1/2) = 30° = π/6 rad
Calculator Tips
Input must be between -1 and 1 for arcsine
Use reverse mode to calculate sine from angle
Results automatically convert between degrees and radians
Principal value is always in [-90°, 90°]
Understanding the Inverse Sine Function (sin⁻¹)
What is sin⁻¹?
The inverse sine function, written as sin⁻¹(x) or arcsin(x), is the inverse of the sine function. It answers the question: "What angle has a sine value of x?" The result is always given as the principal value, which lies between -90° and 90° (-π/2 and π/2 radians).
Domain and Range
- •Domain: [-1, 1] (sine values)
- •Range: [-π/2, π/2] or [-90°, 90°]
- •Only one angle is returned (principal value)
Mathematical Definition
sin⁻¹(x) = y ⟺ sin(y) = x
where -1 ≤ x ≤ 1 and -π/2 ≤ y ≤ π/2
Key Properties
- Odd function: sin⁻¹(-x) = -sin⁻¹(x)
- Continuous: No breaks in the function
- Increasing: Function always rises from left to right
- One-to-one: Each input has exactly one output
Applications
Engineering
Calculate angles in structural analysis, wave analysis, and signal processing
Physics
Determine angles in optics, mechanics, and oscillatory motion problems
Navigation
Calculate bearing angles and elevation angles in GPS and surveying