Arccos Calculator (Inverse Cosine)
Calculate the inverse cosine (arccos) of any value from −1 to 1. Returns the angle in degrees and radians, plus related trig values, quadrant, and unit circle position.
Arcsin Calculator (Inverse Sine)
Calculate the inverse sine (arcsin) of any value from −1 to 1. Returns the angle in degrees and radians, plus cosine, tangent, quadrant, reference angle, and unit circle position.
Range: −1 to 1
Angle (degrees)⧉
30.000000°
Principal value of arcsin in degrees (−90° to 90°)
Angle (radians)⧉
0.523599
Principal value of arcsin in radians (−π/2 to π/2)
Fraction of π⧉
0.166667π
Angle expressed as a multiple of π
Quadrant⧉
I
Arcsin returns Quadrant I (positive) or IV (negative)
Reference Angle⧉
30.000000°
The acute angle to the x-axis
Complementary Angle⧉
60.000000°
90° − θ — this equals arccos(x)
cos(θ)⧉
0.866025
Cosine of the resulting angle — always ≥ 0 for arcsin
tan(θ)⧉
0.577350
Tangent of the resulting angle
Unit Circle Position
Angle
30.0°
sin θ
0.5000
cos θ
0.8660
Common Arcsin Values
| x (input) | Degrees | Radians | Exact |
|---|---|---|---|
| −1 | −90° | −π/2 | sin⁻¹(−1) = −90° |
| −√3/2 ≈ −0.8660 | −60° | −π/3 | sin⁻¹(−√3/2 ≈ −0.8660) = −60° |
| −√2/2 ≈ −0.7071 | −45° | −π/4 | sin⁻¹(−√2/2 ≈ −0.7071) = −45° |
| −0.5 | −30° | −π/6 | sin⁻¹(−0.5) = −30° |
| 0 | 0° | 0 | sin⁻¹(0) = 0° |
| 0.5 | 30° | π/6 | sin⁻¹(0.5) = 30° |
| √2/2 ≈ 0.7071 | 45° | π/4 | sin⁻¹(√2/2 ≈ 0.7071) = 45° |
| √3/2 ≈ 0.8660 | 60° | π/3 | sin⁻¹(√3/2 ≈ 0.8660) = 60° |
| 1 | 90° | π/2 | sin⁻¹(1) = 90° |
Arcsin / Arccos Relationship
| x | arcsin(x) | arccos(x) | Sum (= 90°) |
|---|---|---|---|
| 0 | 0.00° | 90.00° | 90.00° |
| 0.5 | 30.00° | 60.00° | 90.00° |
| 0.7071 | 45.00° | 45.00° | 90.00° |
| 0.866 | 60.00° | 30.00° | 90.00° |
| 1 | 90.00° | 0.00° | 90.00° |
Planning notes, formulas, and examples
About the Arcsin Calculator (Inverse Sine)
The **Arcsin Calculator** computes the inverse sine (sin⁻¹) of a given value, returning the angle whose sine equals your input. Enter any number from −1 to 1, and the tool shows the corresponding angle in degrees and radians, alongside the cosine, tangent, quadrant, reference angle, complementary angle (which equals arccos of the same input), and position on the unit circle.
Arcsine is one of the three fundamental inverse trigonometric functions. It appears in physics when resolving projectile launch angles, in engineering for signal-processing phase calculations, in computer graphics for rotation interpolation, and in geometry whenever you know the ratio of an opposite side to a hypotenuse and need the angle.
The principal branch of arcsin maps [−1, 1] one-to-one onto [−90°, 90°] (−π/2 to π/2 radians). This means the output angle is always in Quadrant I (positive values) or Quadrant IV (negative values). While other angles share the same sine value, the principal value is the standardised answer recognised by calculators and programming languages worldwide.
Visual bars track the angle, sine, and cosine in real time so you can build geometric intuition. A reference table lists the nine most common arcsin values — the same entries you would memorise from a unit-circle chart — and a collapsible section shows the elegant relationship arcsin(x) + arccos(x) = 90°. Eight preset buttons cover every standard angle for instant exploration.
When This Page Helps
Arcsin Calculator (Inverse Sine) helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Angle (degrees), Angle (radians), Fraction of π in one pass.
How to Use the Inputs
- Enter the required inputs (sin⁻¹(x) — Input Value, Angle Units, Decimal Precision).
- Review the output cards, especially Angle (degrees), Angle (radians), Fraction of π, Quadrant.
- Use the common-value presets to compare the result against standard unit-circle angles.
- Compare the result with the formula, diagram, or example values to catch sign, unit, or rounding mistakes.
Formula used
θ = sin⁻¹(x), where −1 ≤ x ≤ 1 and the principal value satisfies −π/2 ≤ θ ≤ π/2 (−90° ≤ θ ≤ 90°). Identity: sin⁻¹(x) + cos⁻¹(x) = 90°. cos(sin⁻¹(x)) = √(1 − x²).Example Calculation
Result: 30°
Using value=0.5, unit=degrees, the calculator returns 30°. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
Tips & Best Practices
- Arcsin always returns a value between −90° and 90° (−π/2 to π/2).
- sin⁻¹(x) + cos⁻¹(x) = 90° for every x in [−1, 1].
- cos(arcsin(x)) = √(1 − x²), which is always ≥ 0 in the principal range.
- For obtuse angles, use the identity: sin(180° − θ) = sin(θ) to find the second solution.
- If your value is outside [−1, 1], no real angle exists — verify your input.
What This Arcsin Calculator (Inverse Sine) Solves
This calculator is tailored to arcsin calculator (inverse sine) workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
How To Interpret The Outputs
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
Study And Practice Strategy
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
Sources & Methodology
Last updated:
Frequently Asked Questions
Arcsin (or sin⁻¹) is the inverse sine function. Given a sine value x, it returns the angle θ such that sin(θ) = x.
To produce a unique output for each input, arcsin is restricted to [−π/2, π/2]. Sine is one-to-one and onto [−1, 1] on this interval.
If θ = arcsin(x) is positive, the obtuse angle is 180° − θ. For example, arcsin(0.5) = 30°, and 180° − 30° = 150° also has sine 0.5.
No. arcsin(x) gives the angle whose sine is x. 1/sin(x) = csc(x), which is the cosecant function.
They are complementary: arcsin(x) + arccos(x) = 90° (π/2 radians) for any x in [−1, 1].
Not for real numbers. The sine function only outputs values in [−1, 1], so its inverse is only defined for inputs in that range.
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