Arcsine Calculator

The arcsine function

The arcsine undoes the sine. If sin θ = x, then arcsin x = θ: you supply the ratio and get back the angle. It is also written asin(x) or sin⁻¹(x), and it is the tool you reach for when you know a side and the hypotenuse of a right triangle and want an angle.

A sine value of 0.5 is produced by 30°, but also by 150°, 390° and infinitely many others. To make the inverse a proper function, arcsine returns only the principal value: the one in the interval from −90° to +90° (−π/2 to +π/2 radians).

• Domain: x must lie between −1 and 1, because no real angle has a sine outside that interval.
• Range: −90° to 90°, or −π/2 to π/2 radians.
• Symmetry: arcsin(−x) = −arcsin(x).
• Link to arccosine: arcsin x + arccos x = 90° for every x in the domain, see the arccosine calculator.

How to use the calculator

Type the sine into the Sine box and press Calculate. You can enter a decimal such as 0.8 or -0.25, or a fraction such as 3/5 or 7.2/8. The result is shown twice, once in degrees and once in radians, each labelled arcsin(x) with your value echoed back (a fraction appears as its decimal value to 14 significant digits).

Values are rounded to six decimals. A whole result is shown bare (30 degrees), a result with one or two decimals gets two (10.50 degrees), three or four decimals are padded to four, and anything longer keeps six, such as 0.523599 radians.

If the number is below −1 or above 1, no result is produced and an error message asks you to correct the value. The fraction format needs a plain numerator and denominator with no spaces around the slash; a fraction with a zero denominator is read as 0, and text that is not a number is read as 0 (or as its leading digits, so 12abc counts as 12).

Arcsine of common values

Finding every angle with a given sine

The calculator gives the angle in the interval −90° to 90°. Because sin(180° − θ) = sin θ, a second angle between 0° and 360° almost always shares the same sine. To list all solutions of sin θ = x, take θ₀ = arcsin x and write:

θ = θ₀ + 360°·k or θ = 180° − θ₀ + 360°·k (k any integer)

Example: sin θ = 0.5 gives θ₀ = 30°, so θ = 30° or 150° within one turn. For sin θ = −0.8, θ₀ = −53.130102°, giving 306.869898° and 233.130102° between 0° and 360°. The only values of x with a single solution per turn are −1 and 1.

Worked examples

A ramp. A ramp raises a load by 3 m over a slope length of 12 m. The sine of the ramp angle is 3 ÷ 12 = 0.25, so the angle is arcsin 0.25 = 14.477512° (0.252680 radians), a little under 14.5°.

A ladder. An 8 m ladder reaches 7.2 m up a wall. Enter 7.2/8, or 0.9, to get arcsin 0.9 = 64.158067° between ladder and ground (1.119770 radians). The 3-4-5 triangle gives another quick check: arcsin(3/5) = 36.869898°.

By hand for small values. For |x| well below 1, a few terms of the series are enough:

arcsin x = x + x³/6 + 3x⁵/40 + 5x⁷/112 + …The result is in radians.

With x = 0.25 the terms are 0.25, 0.002604, 0.000073 and 0.000003, adding up to 0.252680 radian, the same as the calculator. Multiply by 180 ÷ π for degrees.

Common mistakes

• Reading sin⁻¹ as a reciprocal. The −1 exponent marks the inverse function. The reciprocal of sine is the cosecant, 1 ÷ sin θ.
• Forgetting the second angle. Arcsine returns one answer; check whether the geometry allows the supplementary angle 180° − θ, for example in an obtuse triangle.
• Rounded measurements. If a measured ratio comes out marginally above 1, such as 1.0002, the true value is probably 1 and the angle 90°.
• Mixing units. Pick the row, degrees or radians, that matches what your next step expects.

To go from an angle to a ratio, use the sine calculator. More inverse functions, such as arctangent, live in the trigonometry section, and the trigonometry calculator solves complete triangles.