Arcsine Calculator
Enter a value from −1 to 1 to find its arcsine, the angle θ = sin⁻¹(x), in degrees or radians.
Updated
Arcsine Calculator
Our Arcsine Calculator finds the angle whose sine is a given value, arcsin(x) or sin⁻¹(x). Type a decimal, a fraction such as 3/4 or a root such as √3/2 and the angle appears straight away in degrees or radians, to 10 decimal places. Common values give exact answers (arcsin(1/2) = 30° = π/6), and the calculator also checks the result, lists the other angles with the same sine, and explains why values above 1 or below −1 have no arcsine.
Quick answer
The arcsine of x is the angle whose sine is x, between −90° and 90°. For example, arcsin(0.5) = 30° because sin 30° = 0.5. Enter any value from −1 to 1 above and the calculator shows the angle instantly.
- Formula: if sin θ = x, then θ = arcsin(x) = sin⁻¹(x)
- Key values: arcsin(0) = 0°, arcsin(1/2) = 30°, arcsin(√2/2) = 45°, arcsin(1) = 90°
- Domain: −1 ≤ x ≤ 1; any other value has no arcsine
- Range: −90° to 90°, or −π/2 to π/2 radians
What Is Arcsine?
Arcsine is the inverse of the sine function. Sine takes an angle and gives a ratio; arcsine takes the ratio back to the angle. In a right triangle, if you know the side opposite an angle and the hypotenuse, arcsine gives the angle:
θ = arcsin(opposite ÷ hyp)
It is written arcsin x, sin⁻¹ x or asin x. The ⁻¹ means "inverse function", not a power: sin⁻¹ x is not 1 ÷ sin x (that is the cosecant).
Arcsine Table of Common Values
| x | Degrees | Radians |
|---|---|---|
| −1 | −90° | −π/2 |
| −√3/2 ≈ −0.8660 | −60° | −π/3 |
| −√2/2 ≈ −0.7071 | −45° | −π/4 |
| −1/2 | −30° | −π/6 |
| 0 | 0° | 0 |
| 1/2 | 30° | π/6 |
| √2/2 ≈ 0.7071 | 45° | π/4 |
| √3/2 ≈ 0.8660 | 60° | π/3 |
| 1 | 90° | π/2 |
Other values have no simple exact angle, for example arcsin(3/4) ≈ 48.5904° and arcsin(1/3) ≈ 19.4712°. A rounded input such as 0.7071 gives an angle very close to, but not exactly, 45°.
The Arcsine Graph
- Shape: a rising S-curve from (−1, −90°) through (0, 0°) to (1, 90°).
- Always increasing: a bigger x always gives a bigger angle.
- Steep at the ends: near x = ±1 a tiny change in x moves the angle a lot.
- Odd function: arcsin(−x) = −arcsin x, symmetric about the origin.
- Mirror of sine: it is the part of y = sin x between −90° and 90°, reflected in the line y = x.
Why Arcsine Gives One Angle
Many angles share the same sine: sin 30° = sin 150° = sin 390° = 0.5. A function must give one answer, so arcsine returns the principal value, the one between −90° and 90°. To find every angle with sin θ = x, use both of these and add whole turns:
θ = 180° − arcsin(x) + 360° × k
For sin θ = 0.5 between 0° and 360° that gives 30° and 150°. The calculator shows the second angle in its "Other angles" row.
Arcsine Rules and Identities
- Inverse: sin(arcsin x) = x for −1 ≤ x ≤ 1.
- Only one way round: arcsin(sin θ) = θ only for −90° ≤ θ ≤ 90°; arcsin(sin 150°) = 30°.
- Negative values: arcsin(−x) = −arcsin x.
- With arccos: arcsin x + arccos x = 90° (π/2).
- Cosine of arcsine: cos(arcsin x) = √(1 − x2).
- Calculus: the derivative of arcsin x is 1 ÷ √(1 − x2).
Worked Examples
Angle of a ramp. A ramp is 12 m long and rises 1 m. The rise is opposite the angle and the ramp is the hypotenuse:
= arcsin(0.0833)
≈ 4.78°
Light entering water. Light hits water at 45°. By Snell's law, sin r = sin 45° ÷ 1.33:
≈ 0.5317
r = arcsin(0.5317)
≈ 32.12°
A fraction. arcsin(3/4) ≈ 48.5904°, or about 0.8481 rad. The other angle with the same sine is 180° − 48.5904° = 131.4096°.
Arcsine in Excel and Code
| Tool | arcsin(0.5) in degrees |
|---|---|
| Excel / Google Sheets | =DEGREES(ASIN(0.5)) |
| Python | math.degrees(math.asin(0.5)) |
| JavaScript | Math.asin(0.5) * 180 / Math.PI |
| Scientific calculator | DEG mode, then SHIFT sin (sin⁻¹) 0.5 |
ASIN and asin return radians, so convert to degrees as shown. Floating-point maths gives 30.000000000000004 here. Outside −1 to 1, Excel returns #NUM!, Python raises "math domain error" and JavaScript returns NaN.
Common Mistakes
- Reading sin⁻¹ as 1 ÷ sin. sin⁻¹ x is an angle; 1 ÷ sin θ is the cosecant.
- Typing an angle. The input is a sine value (a ratio), not an angle.
- Values above 1. arcsin(1.5) does not exist, because no sine is bigger than 1.
- Forgetting the second angle. sin θ = 0.5 also holds at 150°, not only 30°.
- Wrong angle mode. arcsin(0.5) is 30 in degrees but 0.5236 in radians.
Key Terms
- Inverse
- A function that undoes another; arcsine undoes sine.
- Domain
- The inputs allowed; for arcsine, −1 to 1.
- Range
- The possible outputs; for arcsine, −90° to 90°.
- Principal
- The one angle a function returns out of many that fit.
- Arccos
- The inverse of cosine; arcsin x + arccos x = 90°.
- Cosecant
- 1 ÷ sin θ, often confused with sin⁻¹.
How to Use the Arcsine Calculator
- Enter the value. Type the sine value x, from −1 to 1, or press one of the buttons (1/2, √2/2, √3/2 and 1). You can type a decimal such as 0.75, a fraction such as 3/4, or a root such as √3/2 or sqrt(2)/2.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad) for the answer. The table below the result always shows both.
- Read and copy the result. The angle θ = arcsin(x) appears as you type, to 10 decimal places, with the exact angle for common values (for example arcsin(√2/2) = 45° = π/4 rad). The table also checks the answer and gives the other angle with the same sine. Values above 1 or below −1 have no arcsine and the calculator says so. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter −1/2 and the result is −30°, or −π/6 rad. The table shows that sin(−30°) = −0.5 and that 210° has the same sine.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is arcsine?
Arcsine is the inverse of sine. arcsin(x), also written sin−1(x), is the angle between −90° and 90° whose sine is x. For example arcsin(0.5) = 30°, because sin 30° = 0.5.
What is arcsin 0, arcsin 0.5, arcsin 1 and arcsin(−1)?
arcsin(0) = 0°, arcsin(0.5) = 30° (π/6), arcsin(√2/2) = 45° (π/4), arcsin(√3/2) = 60° (π/3), arcsin(1) = 90° (π/2) and arcsin(−1) = −90° (−π/2).
Why is arcsin(2) undefined?
The sine of an angle is always between −1 and 1, so no angle has a sine of 2. That is why the domain of arcsine is −1 ≤ x ≤ 1. Excel returns #NUM! and Python a math domain error for such values.
What is the range of arcsine?
−90° to 90°, or −π/2 to π/2 radians. Arcsine returns only this principal value, even though other angles have the same sine.
Is sin⁻¹(x) the same as 1/sin(x)?
No. sin−1(x) is the inverse function and returns an angle: sin−1(0.5) = 30°. 1 ÷ sin(x) is the cosecant, csc x: csc 30° = 2.
How do I find all angles with a given sine?
Use θ = arcsin(x) + 360° × k and θ = 180° − arcsin(x) + 360° × k for any whole number k. For sin θ = 0.5 between 0° and 360° that gives 30° and 150°.
What is arcsin(3/4) in degrees?
arcsin(3/4) ≈ 48.5904°, or about 0.8481 radians. It has no simple exact form. The other angle from 0° to 360° with the same sine is 131.4096°.
How do you calculate arcsine in Excel or on a calculator?
In Excel use =DEGREES(ASIN(0.5)) for degrees or =ASIN(0.5) for radians. On a scientific calculator set DEG mode and press SHIFT then sin (the sin−1 key). In code use Math.asin(x) in JavaScript or math.asin(x) in Python, both in radians.
What is the relationship between arcsin and arccos?
For any x from −1 to 1, arcsin x + arccos x = 90° (π/2). So arccos(0.5) = 90° − 30° = 60°.
Is the Arcsine Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Arcsine Calculator to find the angle for any sine value in degrees or radians instantly.