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.
| Function | Result (°) | Domain | Status |
|---|---|---|---|
| arccos(x) | 60.000000° | [−1, 1] | ✓ Valid |
| arcsin(x) | 30.000000° | [−1, 1] | ✓ Valid |
| arctan(x) | 26.565051° | (−∞, ∞) | ✓ Valid |
| arcsec(x) | — | |x| ≥ 1 | ✗ Outside |
| arccsc(x) | — | |x| ≥ 1 | ✗ Outside |
| arccot(x) | 63.434949° | x ≠ 0 | ✓ Valid |
| x | Degrees | Exact Radians | Decimal Radians |
|---|---|---|---|
| 1 | 0° | 0 | 0 |
| √3/2 ≈ 0.8660 | 30° | π/6 | 0.5236 |
| √2/2 ≈ 0.7071 | 45° | π/4 | 0.7854 |
| 1/2 = 0.5 | 60° | π/3 | 1.0472 |
| 0 | 90° | π/2 | 1.5708 |
| −1/2 = −0.5 | 120° | 2π/3 | 2.0944 |
| −√2/2 ≈ −0.7071 | 135° | 3π/4 | 2.3562 |
| −√3/2 ≈ −0.8660 | 150° | 5π/6 | 2.6180 |
| −1 | 180° | π | 3.1416 |
| Property | Value |
|---|---|
| Function | y = arccos(x) = cos⁻¹(x) |
| Domain | [−1, 1] |
| Range | [0, π] or [0°, 180°] |
| Identity | arccos(x) + arcsin(x) = π/2 |
| Symmetry | arccos(−x) = π − arccos(x) |
| Derivative | d/dx arccos(x) = −1/√(1 − x²) |
| At x = 0 | arccos(0) = π/2 = 90° |
| At x = 1 | arccos(1) = 0 |
| At x = −1 | arccos(−1) = π = 180° |
The **Inverse Cosine (arccos) Calculator** computes the angle whose cosine equals a given value. Enter any number between −1 and 1, and the tool returns the result in degrees, radians, gradians, and turns — plus a full set of related inverse trigonometric values at the same input.
The arccos function is fundamental in geometry, physics, and engineering. In geometry, it recovers an angle from a known adjacent-to-hypotenuse ratio. In physics, it finds the angle between two vectors using the dot-product formula θ = arccos(a·b / |a||b|). In computer graphics, arccos drives lighting calculations, reflection angles, and camera field-of-view computations.
This calculator goes beyond a single result. It shows all six inverse trig functions evaluated at your input, highlights which are valid (inside their domain) and which are not, and verifies the result by computing cos(arccos(x)) as a round-trip check. A visual bar tracks both the input position across [−1, 1] and the output angle across [0°, 180°].
Eight preset buttons cover the standard unit-circle values — 0°, 30°, 45°, 60°, 90°, 120°, 135°, and 180° — so you can load any textbook answer quickly. A fraction mode lets you enter exact ratios like 1/2 or −√3/2 without rounding first. Common values and domain/range reference tables provide a quick lookup for students and professionals alike.
Inverse Cosine (arccos) Calculator — Degrees, Radians & More 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 Degrees, Radians, Gradians in one pass.
θ = arccos(x), where x ∈ [−1, 1] and θ ∈ [0, π]. In degrees: θ° = arccos(x) × 180/π. Identity: arccos(x) + arcsin(x) = π/2. Symmetry: arccos(−x) = π − arccos(x). Derivative: d/dx arccos(x) = −1/√(1 − x²).Result: 60°
Using value=0.5, unit=degrees, the calculator returns 60°. 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 cosine (arccos) calculator — degrees, radians & more 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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The domain is [−1, 1]. Cosine only produces values in this range, so only these values can be "reversed." Inputs outside [−1, 1] yield no real angle.
The range is [0, π] in radians, or [0°, 180°] in degrees. This is the principal value range chosen so that arccos is a proper (single-valued) function.
They are complementary: arccos(x) + arcsin(x) = π/2 (90°) for all x in [−1, 1]. Knowing one immediately gives the other.
Arccos returns only the principal value (0° to 180°). The equation cos θ = x has infinitely many solutions: θ = ±arccos(x) + 360°n. Your problem may require a different branch.
This calculator handles real inputs in [−1, 1]. For complex arguments, arccos is extended using the formula arccos(z) = −i·ln(z + i√(1 − z²)), which returns complex angles.
The most common use is finding the angle between two vectors: θ = arccos(a·b / |a||b|). It appears in optics (refraction angles), orbital mechanics, and electromagnetism.
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 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.
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.