What is the inverse sine of x?
Type a number x from −1 to 1. The inverse sine calculator shows arcsin x, the angle whose sine is x, in degrees and radians, and the exact angle for values such as 1/2 and √2/2.
- arcsin x
- 30°
arcsin(0.5) = 30°.
- In radians
- 0.523599
- Exact angle
- π/6 = 30° (sin 30° = 1/2)
arcsin x: 30°. arcsin(0.5) = 30°.
How to calculate
Finds arcsin x, the angle from −90° to 90° whose sine is x, in degrees and radians, exact as a multiple of π for the special values.
Example with the default inputs (Value x 0.5): arcsin(0.5) = 30°.
Method: arcsin x is the angle θ from −90° to 90° (−π/2 to π/2) with sin θ = x. When x is within 10⁻¹⁴ of the sine of a multiple of 15° in that range, the angle is that multiple exactly; otherwise it is computed in 64-bit floating point.
- x is from −1 to 1; the sine of a real angle is never outside that range.
- The answer is the principal value. Other angles with the same sine are 180° − θ and those angles plus whole turns.
Worked examples
Each example is checked against the calculator on every build.
- Value x 0.5 gives arcsin x 30°, In radians 0.523599, Exact angle π/6 = 30° (sin 30° = 1/2).Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin: domain [−1, 1], range [−π/2, π/2]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions
- Value x -1 gives arcsin x -90°, Exact angle -π/2 = -90° (sin -90° = -1).Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin: domain [−1, 1], range [−π/2, π/2]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions
- Value x 0.707107 gives arcsin x 45°, Exact angle π/4 = 45° (sin 45° = √2/2).Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin: domain [−1, 1], range [−π/2, π/2]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions
- Value x 0.3 gives arcsin x 17.457603°, In radians 0.304693.Source: NIST DLMF §4.23 (inverse trigonometric functions, principal values), https://dlmf.nist.gov/4.23
How it works
arcsin x is the angle θ with sin θ = x and −90° ≤ θ ≤ 90° (−π/2 ≤ θ ≤ π/2). This range holds exactly one angle for each x from −1 to 1.
Exact angles. The calculator checks the sines of the multiples of 15° from −90° to 90°: 0, ±(√6 − √2)/4, ±1/2, ±√2/2, ±√3/2, ±(√6 + √2)/4 and ±1. If x is within 10⁻¹⁴ of one of them, the angle is that multiple of 15° exactly, and the page shows it as a multiple of π. Otherwise the angle is computed in 64-bit floating point.
Output format. The angle shows in degrees. "In radians" shows the same angle in radians as a decimal. The exact angle shows as text, such as π/6 = 30° (sin 30° = 1/2), with - for negative values.
Assumptions
- x is from −1 to 1. A value outside that range is not accepted, because no real angle has that sine.
- The answer is the principal value; the FAQ shows how to find the other angles.
Worked examples by hand
x = 0.5. sin 30° = 1/2 and 30° is between −90° and 90°, so arcsin 0.5 = 30° = π/6 ≈ 0.5235987755982988 rad.
x = −1. sin(−90°) = −1, so arcsin(−1) = −90° = −π/2 ≈ −1.5707963267948966 rad.
x = 0.7071067811865476 (√2/2). sin 45° = √2/2, so arcsin x = 45° = π/4 ≈ 0.7853981633974483 rad.
x = 0.3. 0.3 is not a special value, so arcsin 0.3 ≈ 0.30469265401539747 rad ≈ 17.4576°.
Other questions people ask
What is inverse sine?
Inverse sine, written arcsin x or sin⁻¹ x, undoes the sine: it gives the angle whose sine is x. Because many angles share a sine, it gives the principal value, the one from −90° to 90°.
What is arcsin 0.5?
arcsin 0.5 = 30° = π/6 radians, because sin 30° = 1/2. Other angles with sine 1/2, such as 150°, are not principal values.
Why does arcsin 2 give an error?
The sine of a real angle is always from −1 to 1, so no angle has sine 2. The calculator accepts x from −1 to 1 only.
Is sin⁻¹ x the same as 1 ÷ sin x?
No. sin⁻¹ x is the inverse function, arcsin x. The reciprocal 1 ÷ sin x is the cosecant, csc x.
How do I find all the angles with a given sine?
Take θ = arcsin x. The angles with sine x are θ + 360°k and 180° − θ + 360°k for any whole number k. For x = 1/2: 30°, 150°, 390°, 510°, and so on.
How precise is the answer?
For the special values (0, ±1/2, ±√2/2, ±√3/2, ±1 and the 15° values) the angle is exact. Typing √2/2 as 0.7071067811865476 counts, because it is within 10⁻¹⁴ of √2/2. Other values are computed to about 16 significant digits.