Arccos Calculator

Compute the arccosine of any value between -1 and 1 — the inverse of the cosine function. Get results in both radians and degrees, visualize the principal branch of y = arccos(x), and explore fundamental properties, real‑world applications, and historical context.

Arccos(x) returns the angle θ ∈ [0, π] such that cos(θ) = x.
arccos(-1) = 180° arccos(-√3/2) = 150° arccos(-0.5) = 120° arccos(0) = 90° arccos(0.5) = 60° arccos(√2/2) = 45° arccos(√3/2) = 30° arccos(1) = 0°
Arccos Result & Properties
arccos(0.5) =
1.0472 radians
60.00 degrees
Principal value range
θ ∈ [0, π]   (0° ≤ θ ≤ 180°)
✔ valid input within domain
Inverse property: cos(arccos(x)) = x   for all x ∈ [-1,1] ;   arccos(cos(θ)) = θ only if θ ∈ [0, π].
Graph of y = arccos(x) (principal branch)
y = arccos(x)
Current point (x₀, arccos(x₀))
Endpoint boundaries

Domain: [-1,1]   |   Range: [0, π] (≈ 0 to 3.1416 rad)

Privacy-first computation: All calculations happen locally in your browser. No data is transmitted or stored.

Understanding Arccos (Inverse Cosine): definition, derivation & significance

The arccosine function, denoted arccos(x) or cos⁻¹(x), is the inverse of the cosine function restricted to the interval [0, π]. For a given value x between -1 and 1, arccos(x) returns the unique angle θ in that interval whose cosine equals x. This function is fundamental in geometry, physics, and engineering, particularly for determining angles from side ratios in right triangles and for solving trigonometric equations.

For −1 ≤ x ≤ 1,   θ = arccos(x)   ⇔   cos(θ) = x   and   θ ∈ [0, π].

The derivative: d/dx arccos(x) = −1/√(1−x²), |x| < 1. This negative derivative indicates the function is strictly decreasing.

Historical & mathematical roots

Inverse trigonometric functions were formalized in the 17th and 18th centuries alongside calculus. The notation "arccos" (arcus cosinus) emphasizes that the result is an arc (angle) whose cosine is the given value. Leonhard Euler and other mathematicians extensively used these functions. Today, arccos appears in coordinate transformations, signal processing (phase unwrapping), and in the law of cosines to compute angles of a triangle from side lengths.

Key properties and identities

  • Symmetry: arccos(−x) = π − arccos(x) (reflection property).
  • Complementary angle: arccos(x) + arcsin(x) = π/2 for all x ∈ [−1,1].
  • Relationship with arctan: arccos(x) = arctan(√(1−x²)/x) for x > 0, with appropriate quadrant adjustments.
  • Derivative and integral: ∫ arccos(x) dx = x arccos(x) − √(1−x²) + C.
  • Series expansion: arccos(x) = π/2 − (x + (1/6)x³ + (3/40)x⁵ + …) for |x| ≤ 1.

Practical applications across disciplines

Geometry & Triangulation

Given three sides of a triangle, the law of cosines uses arccos to compute any angle. Essential in geodesy, robotics (inverse kinematics), and computer graphics (angle between vectors).

Signal Processing

Arccos is used in phase detection, FM demodulation, and in the computation of the instantaneous phase of analytic signals.

Navigation & Astronomy

The haversine formula for great-circle distance between two points on a sphere involves arccos of the central angle. Also used in celestial navigation to compute altitude or azimuth.

Step-by-step calculation method

  1. Ensure input x lies in the closed interval [−1,1]. If not, arccos is undefined in reals.
  2. Use the built‑in Math.acos function (IEEE 754 compliant) returning value in radians between 0 and π.
  3. Convert to degrees by multiplying by 180/π.
  4. Validate with known identities: arccos(0) = π/2, arccos(1) = 0, arccos(-1) = π.

Our calculator implements double‑precision floating‑point arithmetic, ensuring accuracy to 15 decimal digits. The interactive graph shows the exact point (x₀, arccos(x₀)) on the principal branch, reinforcing the geometric interpretation: arccos returns the angle on the unit circle (from 0 to π) whose horizontal coordinate equals x₀.

Common mistakes and misconceptions

  • Confusing arccos with 1/cos: arccos(x) is not the reciprocal; sec(x) = 1/cos(x) is different.
  • Domain errors: arccos(1.5) is not a real number. The calculator warns for out‑of‑range inputs.
  • Principal value nuance: arccos(cos(300°)) ≠ 300°; it returns 60° because the result must be within [0°, 180°].
  • Decreasing nature: arccos(x) decreases as x increases, opposite to arcsin.

Frequently Asked Questions

They are identical notations. cos⁻¹(x) represents the inverse function, not the multiplicative inverse. For reciprocal, use sec(x).

Cosine is decreasing on [0, π] and covers all values from 1 to −1 uniquely. This interval is the conventional principal branch, giving a one‑to‑one mapping.

Very accurate: arccos(1) gives exactly 0, arccos(-1) gives π (~3.141592653589793), within machine epsilon.

arccos(x) equals the angle (in radians) whose cosine is x. On the unit circle, that angle's horizontal coordinate matches x, and the angle lies between 0 and π (upper half-plane).

Yes. Given three sides a, b, c of a triangle, angle C opposite side c is arccos((a²+b²−c²)/(2ab)). This is the law of cosines.
Authoritative references: Wolfram MathWorld: Inverse Cosine | Khan Academy: Intro to arccos | Abramowitz, M. and Stegun, I.A. "Handbook of Mathematical Functions" (1972).

Expert verification: This tool implements the standard IEEE-754 arccosine function. All content has been reviewed by getzenquery tech team. Last update: May 2026.