Use this calculator to find the arcsine (inverse sine) of a number between -1 and 1.
Formula:
Arcsin(x) = sin⁻¹(x)Welcome to the Easy Converters Arcsin Calculator – your go-to tool for calculating the inverse sine (also known as arcsin or sin⁻¹) of any real number between –1 and 1. Whether you’re a student learning trigonometry, an engineer solving wave equations, or a developer working with angles, this free calculator provides fast, accurate results in both degrees and radians.
The **arcsin** or **inverse sine function** is the inverse of the sine function. It answers the question: "What angle has a given sine value?" In simple terms, if:
sin(θ) = x then arcsin(x) = θ
Example: sin(30°) = 0.5 ⇒ arcsin(0.5) = 30°
The inverse sine function is written as:
θ = arcsin(x)
where –1 ≤ x ≤ 1
This is equivalent to solving for the angle in the sine equation.
| Value (x) | arcsin(x) in Degrees | arcsin(x) in Radians |
|---|---|---|
| 1 | 90° | π/2 |
| 0.866 | 60° | π/3 |
| 0.707 | 45° | π/4 |
| 0.5 | 30° | π/6 |
| 0 | 0° | 0 |
| –0.5 | –30° | –π/6 |
| –1 | –90° | –π/2 |
| Sine Function (sin) | Inverse Sine (arcsin) |
|---|---|
| Input is an angle | Input is a ratio |
| Output is a ratio (between –1 and 1) | Output is an angle |
| Many-to-one function | One-to-one inverse function |
The **unit circle** is a great tool to understand inverse trigonometric functions. On the unit circle:
The y-coordinate represents the sine of the angle.
So, to find arcsin(x), you’re essentially asking: "Which angle gives a y-coordinate of x?"
Math.asin(x) in JavaScriptmath.asin(x) in Pythonasin(x) in C/C++ using math libraries
These functions return the angle in radians, so if you need degrees, use:
degrees = radians × (180 / π)
In any right triangle, if you know the **opposite** side and the **hypotenuse**, you can find the angle using:
θ = arcsin(opposite / hypotenuse)
This is especially useful in real-world surveying, structural design, and construction calculations.
Whether you're on a desktop, tablet, or mobile device, the Arcsin Calculator is fully responsive and optimized for fast performance. Best of all – no popups, no distractions, and no login required.
Q: What is the range of arcsin?
A: The range of arcsin is from –π/2 to π/2 (–90° to 90°).
Q: Is arcsin the same as sin⁻¹?
A: Yes. Sin⁻¹(x) and arcsin(x) both mean inverse sine.
Q: Can I enter values outside –1 to 1?
A: No, the sine function only returns values within that range, so arcsin is only defined for –1 ≤ x ≤ 1.
The Easy Converters Arcsin Calculator is the perfect tool to solve inverse sine values quickly and correctly. Whether you're preparing for exams, writing software, or solving real-world problems, this tool makes trigonometry simple. With clear explanations, fast performance, and mobile-friendly design, you’ll get results you can trust – every time.
To truly understand arcsin, it’s helpful to consider how the regular sine function behaves. The sine function takes an angle and returns the ratio of the opposite side to the hypotenuse in a right triangle. This ratio is always between –1 and 1, which is why the domain of arcsin is limited to that same interval.
When you apply the inverse sine function, you’re asking: “Which angle has this sine ratio?” The answer is an angle between –90° and 90° (or –π/2 and π/2 radians), because this is where the sine function is one-to-one and can be inverted reliably.
The arcsin function is a smooth, increasing curve. Here are some key properties of the arcsin graph:
These features make arcsin predictable and useful in mathematical modeling and analysis.
sin(arcsin(x)) = xarcsin(sin(x)) = x (for –π/2 ≤ x ≤ π/2)arcsin(x) + arccos(x) = π/2arcsin(–x) = –arcsin(x)These identities are useful in simplifying trigonometric expressions and solving equations involving multiple trig functions.
Arcsin plays an important role in:
Example:
d/dx [arcsin(x)] = 1 / √(1 – x²)
Arcsin is widely used in programming, especially for simulations, geometry, and data processing. Examples:
Always remember: programming languages return the result of arcsin in radians. If you want degrees, use conversion formulas.
Suppose you know the length of the opposite side and hypotenuse. You can find the angle using:
θ = arcsin(opposite / hypotenuse)
Example:
Opposite = 5, Hypotenuse = 10
θ = arcsin(5/10) = arcsin(0.5) = 30°
This use-case appears often in architecture, surveying, and physics problems.
radians = degrees × (π / 180)degrees = radians × (180 / π)This conversion is essential if you’re switching between calculators, programming environments, or academic references.
For small values of x, arcsin(x) can be approximated using a Taylor Series:
arcsin(x) ≈ x + x³/6 + 3x⁵/40 + 5x⁷/112 + …
While this is not required for most users, it gives a deep insight into how arcsin can be calculated numerically without a calculator.
The Arcsin Calculator by Easy Converters is the ideal tool for students, teachers, developers, and professionals. It saves time, avoids error-prone manual work, and delivers precise results in both radians and degrees. Whether you’re calculating angles in triangles, analyzing waveforms, or coding game mechanics, this tool brings you one step closer to mastering trigonometry.
Use it anywhere, anytime – ad-free, clutter-free, and always accurate.