Inverse Sine Calculator
Calculate arcsin(x) - the inverse sine function for values between -1 and 1
Calculate Inverse Sine
Domain: [-1, 1]
Results
In Degrees
In Radians
Explanation
sin(30°) = 1/2, therefore arcsin(1/2) = 30°
Special Values of Inverse Sine
| Input (x) | arcsin(x) in Degrees | arcsin(x) in Radians | Exact Value |
|---|---|---|---|
| -1 | -90° | -π/2 | -π/2 ≈ -1.5708 |
| -√3/2 | -60° | -π/3 | -π/3 ≈ -1.0472 |
| -√2/2 | -45° | -π/4 | -π/4 ≈ -0.7854 |
| -1/2 | -30° | -π/6 | -π/6 ≈ -0.5236 |
| 0 | 0° | 0 | 0 |
| 1/2 | 30° | π/6 | π/6 ≈ 0.5236 |
| √2/2 | 45° | π/4 | π/4 ≈ 0.7854 |
| √3/2 | 60° | π/3 | π/3 ≈ 1.0472 |
| 1 | 90° | π/2 | π/2 ≈ 1.5708 |
Quick Reference
Properties
Understanding the Inverse Sine Function
What is Inverse Sine?
The inverse sine function, denoted as arcsin(x) or sin⁻¹(x), is the inverse function of the sine. It takes a value between -1 and 1 and returns the angle whose sine equals that value.
Key Properties
- •Domain: [-1, 1] - input must be between -1 and 1
- •Range: [-π/2, π/2] radians or [-90°, 90°]
- •The function is increasing (as x increases, arcsin(x) increases)
- •arcsin(x) is an odd function: arcsin(-x) = -arcsin(x)
Applications
The inverse sine function is used in trigonometry, engineering, physics, and computer graphics for finding angles when the sine value is known.
Common Use Cases
- •Finding angles in right triangles
- •Solving trigonometric equations
- •Wave analysis and signal processing
- •Physics calculations involving oscillations
Key Formulas
Basic Definition
Relationships
Step-by-Step Examples
Example 1: Find arcsin(1/2)
Step 1: We need to find the angle whose sine is 1/2
Step 2: From the unit circle, we know sin(30°) = 1/2
Step 3: Since 30° is in the range [-90°, 90°], arcsin(1/2) = 30°
Answer: arcsin(1/2) = 30° = π/6 radians
Example 2: Find arcsin(-√2/2)
Step 1: We need to find the angle whose sine is -√2/2
Step 2: From the unit circle, we know sin(-45°) = -√2/2
Step 3: Since -45° is in the range [-90°, 90°], arcsin(-√2/2) = -45°
Answer: arcsin(-√2/2) = -45° = -π/4 radians