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Inverse Sin Calculator | Find Arcsine in Degrees or Radians

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Convert a sine ratio to its angle with arcsin, in degrees or radians, including the second solution in the full circle. Shows the principal range — ideal for trigonometry, physics and engineering problems.

Last updated: June 2026 · Free · No sign-up required

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Results

Enter values above and click Calculate to see your result instantly.

Quick reference

sin valuearcsin (degrees)arcsin (radians)
00°0
0.530°π/6
0.707145°π/4
0.866060°π/3
190°π/2
−0.5−30°−π/6

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

calculate Accurate to the spec

Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

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Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

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Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

Input valuePrincipal angle
0 to 10° to 90° (Quadrant I)
−1 to 0−90° to 0° (Quadrant IV)
Outside [−1, 1]Undefined — sine never exceeds 1

lightbulb Worked example: ramp incline from rise and run

A 3 m ramp rises 1.2 m. What is its angle? sin θ = opposite/hypotenuse = 1.2/3 = 0.4, so θ = arcsin(0.4) ≈ 23.578°. In radians: 23.578 × π/180 ≈ 0.4115. Second solution: 180° − 23.578° = 156.42°.

Result: ≈ 23.58° (≈ 0.412 rad); second solution 156.42°.

What this means: The 23.58° answer is the physical ramp — about a 43.6% grade (tan 23.58° ≈ 0.436), far steeper than an accessible ramp. The 156.42° solution satisfies the equation but describes no real ramp, so the principal value is the one that matters.

Formula & methodology

Formula: θ = arcsin(x), −90° ≤ θ ≤ 90°; second solution in [0°, 360°): 180° − θ

How arcsine is computed

Arcsine inverts the sine function: θ = arcsin(x) is the angle whose sine equals x, defined only for −1 ≤ x ≤ 1.

θ = arcsin(x), principal range −90° ≤ θ ≤ 90°

  • Results are given in both degrees and radians (radians = degrees × π/180).
  • Because sin(θ) = sin(180° − θ), a second solution exists: 180° − θ in [0°, 360°).
  • Inputs outside [−1, 1] have no real arcsine and are rejected.

Reading the result

arcsin(0.5) = 30° with second solution 150°. For a ramp rising 1.2 m over a 3 m slope, sin θ = 0.4 gives θ ≈ 23.58° — the supplementary 156.42° is mathematically valid but not a physical ramp angle.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is arcsin of 0.5?
The Inverse Sine Calculator | Arcsin Angle Tool is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Finds the inverse sine (arcsine) of any value between −1 and 1. Enter a number and choose degrees or radians, and the tool returns the angle whose sine equals that value, along with the principal-angle working. Useful for trigonometry homework, physics There is no limit on usage, and it works on any device.
How do I do inverse sine on a calculator?
Enter the required values into the input fields and press the calculate button. The tool applies the formula θ = arcsin(x), −90° ≤ θ ≤ 90°; second solution in [0°, 360°): 180° − θ and returns the result with a short explanation of each step, so you can check the working yourself.
Why is arcsin undefined for values over 1?
Because the sine of any real angle is always between −1 and 1 — there is no real angle whose sine exceeds 1. The domain of arcsin is therefore restricted to [−1, 1], and inputs outside that range have no real solution. The calculator flags these inputs rather than returning an error, and for complex analysis the results exist but are outside real trigonometry.
What range does arcsin return?
The Inverse Sine Calculator | Arcsin Angle Tool is a free online tool that finds the inverse sine (arcsine) of any value between −1 and 1. enter a number and choose degrees or radians, and the tool returns the angle whose sine equals that value, along with the principal-angle working. useful for trigonometry homework, physics. You enter a few values, and it returns an accurate result together with a step-by-step explanation, so it works as both a calculator and a quick reference for the underlying concept.
What's the difference between arcsin and cosecant?
They are completely different functions with confusingly similar names. Arcsin (sin⁻¹) answers 'what angle has this sine?' — arcsin(0.5) = 30°. Cosecant (csc) is the reciprocal of the sine VALUE: csc(30°) = 1 ÷ 0.5 = 2. The superscript −1 in sin⁻¹ means inverse function, not reciprocal. The calculator applies arcsin, the inverse function.
What angle gives a sine of 0.5?
arcsin(0.5) = 30° (or π/6 ≈ 0.5236 radians) — the classic 30-60-90 triangle ratio. Because sine is positive in quadrants I and II, there is a second solution at 180° − 30° = 150°. The calculator lists both the principal angle and the supplementary solution.
How steep is a ramp that rises 1.2 m over 3 m?
sin θ = opposite/hypotenuse = 1.2/3 = 0.4, so θ = arcsin(0.4) ≈ 23.58°. That is a steep ramp — about a 43.6% grade. The 156.42° second solution is geometrically valid but not a physical ramp angle.
Why does arcsin only accept values from −1 to 1?
Because sine itself only outputs values in that range — no angle has a sine of 1.5, so there is nothing to invert. Asking for arcsin(2) is like asking which real number squared gives −1: the answer does not exist. The calculator rejects out-of-range inputs rather than guessing.
What is the difference between arcsin(x) and 1/sin(x)?
arcsin(x) is the inverse function — the angle whose sine is x. 1/sin(x) is the reciprocal, called cosecant (csc). They are completely different: arcsin(0.5) = 30°, while 1/sin(30°) = 2. Calculator buttons labeled sin⁻¹ mean arcsin, not the reciprocal.

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