Calculate all six inverse trig functions simultaneously: arcsin, arccos, arctan, arccsc, arcsec, arccot. Enter a value and see every result with domain checking, identities, and visual comparison.
| Function | Domain | Range (rad) | Range (deg) |
|---|---|---|---|
| arcsin(x) | [−1, 1] | [−π/2, π/2] | [−90°, 90°] |
| arccos(x) | [−1, 1] | [0, π] | [0°, 180°] |
| arctan(x) | (−∞, ∞) | (−π/2, π/2) | (−90°, 90°) |
| arccsc(x) | |x| ≥ 1 | [−π/2, π/2]\{0} | [−90°, 90°]\{0°} |
| arcsec(x) | |x| ≥ 1 | [0, π]\{π/2} | [0°, 180°]\{90°} |
| arccot(x) | (−∞, ∞) | (0, π) | (0°, 180°) |
| Identity | Condition |
|---|---|
| arcsin(x) + arccos(x) = π/2 | For all x ∈ [−1, 1] |
| arctan(x) + arccot(x) = π/2 | For x > 0 |
| arccsc(x) = arcsin(1/x) | For |x| ≥ 1 |
| arcsec(x) = arccos(1/x) | For |x| ≥ 1 |
| arccot(x) = arctan(1/x) | For x > 0 |
| arcsin(−x) = −arcsin(x) | Odd function |
| arccos(−x) = π − arccos(x) | Symmetry about x = 0 |
| arctan(−x) = −arctan(x) | Odd function |
| sin(arcsin(x)) = x | Round-trip for x ∈ [−1, 1] |
| arctan(x) = arcsin(x/√(1+x²)) | Conversion identity |
| Function | Defined As | Domain Restriction |
|---|---|---|
| arccsc(x) | arcsin(1/x) | |x| ≥ 1 (because sin ∈ [−1,1]) |
| arcsec(x) | arccos(1/x) | |x| ≥ 1 (because cos ∈ [−1,1]) |
| arccot(x) | arctan(1/x) adjusted | All reals (x ≠ 0; at 0, arccot = π/2) |
The **Inverse Trigonometric Functions Calculator** evaluates all six inverse trig functions — arcsin, arccos, arctan, arccsc, arcsec, and arccot — for any input value, displaying every result simultaneously. Enter a number, and see which functions accept it (domain check), what angles they return in degrees and radians, and how the results relate through fundamental identities.
Inverse trigonometric functions reverse the basic trig functions: given a ratio, they return the corresponding angle. They are essential in calculus (antiderivatives of rational functions), physics (resolving angles from measured ratios), engineering (signal phase analysis), and computer graphics (lighting and rotation calculations). Understanding all six together — their domains, ranges, and interconnections — is key to mastering trigonometry.
This calculator offers two usage modes: evaluate all six functions at once for a single input, or focus on one specific function. A visual bar chart compares results across all six functions, with hatched bars for out-of-domain entries. The domain/range reference table highlights which functions can accept your current input, and a comprehensive identities table lists the ten most important inverse trig identities with their conditions.
Eight presets span the typical range from −1 to 2, covering values that exercise every domain boundary. The precision control lets you set up to 12 decimal places, and the unit selector toggles between degrees, radians, or both.
Inverse Trigonometric Functions Calculator — All 6 at Once 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 arcsin + arccos, arctan + arccot in one pass.
arcsin(x): domain [−1,1], range [−π/2, π/2]. arccos(x): domain [−1,1], range [0, π]. arctan(x): domain ℝ, range (−π/2, π/2). arccsc(x) = arcsin(1/x) for |x|≥1. arcsec(x) = arccos(1/x) for |x|≥1. arccot(x) = arctan(1/x) adjusted for sign. Key identity: arcsin(x) + arccos(x) = π/2.Result: Computed from the entered values
Using v=0, the calculator returns Computed from the entered values. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to inverse trigonometric functions calculator — all 6 at once 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.
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.
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.
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They are arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹), arccsc (csc⁻¹), arcsec (sec⁻¹), and arccot (cot⁻¹). Each reverses the corresponding trig function, returning an angle from a ratio.
Each inverse trig function has a restricted domain. arcsin and arccos only accept [−1, 1]; arccsc and arcsec only accept |x| ≥ 1. If your input falls outside a function's domain, no real angle exists.
Through reciprocal identities: arccsc(x) = arcsin(1/x), arcsec(x) = arccos(1/x), and arccot(x) = arctan(1/x) (with sign adjustment for negative x).
Because sin and cos are co-functions: sin(θ) = cos(90° − θ). So if arcsin(x) = θ, then cos(90° − θ) = x, meaning arccos(x) = 90° − θ.
arcsin, arctan, and arccsc can return negative values (their ranges include negative angles). arccos, arcsec, and arccot return only non-negative angles in their principal ranges.
arctan, because ∫ 1/(1+x²) dx = arctan(x) + C appears in many standard integrals. arcsin appears in ∫ 1/√(1−x²) dx = arcsin(x) + C.
Calculate the inverse sine (arcsin) of any value. Get results in degrees, radians, gradians, and turns with domain checking, unit circle visualization, and all 6 inverse trig comparisons.
Calculate the inverse cosine (arccos) of any value. Get results in degrees, radians, gradians, and turns with domain checking, related trig values, and a common values reference table.
Calculate arctan(x) or atan2(y,x) with full quadrant support. Get results in degrees, radians, gradians, and turns with quadrant identification, common values table, and atan-vs-atan2 comparison.