Arccos Calculator

Definition

The inverse cosine, written arccos(x) or acos(x), is the inverse of the restriction of cosine to an interval on which cosine is one-to-one. The map cos : ℝ → [−1, 1] is periodic and is not injective on the whole real line: cos(θ) = cos(−θ) = cos(θ + 2πk) for integers k. Hence no global single-valued inverse exists. On the closed interval [0, π], cos is continuous and strictly decreasing from 1 to −1, so it is bijective onto [−1, 1]. The inverse of that restriction is arccos : [−1, 1] → [0, π].

The range [0, π] is the principal value. It is the standard branch in the NIST Digital Library of Mathematical Functions, §4.23, and in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 4. In degrees the principal range is [0°, 180°]. This calculator uses the same branch: it computes arccos in radians and, when requested, converts α° = α · (180/π).

Historical and conceptual context

Inverse trigonometric functions arose wherever an angle must be recovered from a sine, cosine, or tangent value — in navigation, astronomy, and calculus. In modern analysis arccos is one of the six standard inverse trigonometric functions. MathWorld treats Inverse Cosine as multi-valued in the complex plane, with a distinguished real branch on [−1, 1].

Conceptually, arccos does not select an angle outside the principal branch. For x = 0 both 90° and 270° have cosine zero, yet arccos(0) = π/2. That choice is an analytic convention, not a physical ban on other angles. When a two-argument angle from (cos, sin) with quadrant resolution is required, atan2 is used rather than arccos alone.

Formal properties

Domain: x ∈ [−1, 1]. A number outside this interval is not a real cosine value, so no real result exists. Codomain (principal range): [0, π]. The function arccos is continuous on [−1, 1], strictly decreasing, with arccos(1) = 0, arccos(0) = π/2, arccos(−1) = π.

Identities for x ∈ [−1, 1]: cos(arccos(x)) = x; arccos(cos θ) = θ for θ ∈ [0, π]; arccos(x) + arcsin(x) = π/2; arccos(−x) = π − arccos(x); sin(arccos(x)) = √(1 − x²) ≥ 0 on the principal branch.

Derivative: for |x| < 1, (arccos x)′ = −1/√(1 − x²). At the endpoints the derivative has vertical asymptotes. Antiderivative: ∫ arccos(x) dx = x·arccos(x) − √(1 − x²) + C.

Full derivation of the branch and the derivative

Step 1. Let f(θ) = cos θ on [0, π]. Then f′(θ) = −sin θ ≤ 0 and is strictly negative on (0, π), so f is strictly decreasing. Since f(0) = 1 and f(π) = −1, the image is [−1, 1].

Step 2. Therefore a unique inverse g : [−1, 1] → [0, π] exists. Write g = arccos. By definition, cos(arccos(x)) = x and arccos(cos θ) = θ on the indicated sets.

Step 3 (derivative). Let y = arccos(x), so x = cos y. Differentiating in y yields dx/dy = −sin y, hence dy/dx = −1/sin y. On (0, π), sin y > 0 and sin y = √(1 − x²), so (arccos x)′ = −1/√(1 − x²).

Step 4 (link to arcsin). Let α = arcsin(x) ∈ [−π/2, π/2]. Then cos(π/2 − α) = x and π/2 − α ∈ [0, π]. Uniqueness of arccos gives arccos(x) = π/2 − arcsin(x).

Special cases

Standard values: arccos(1) = 0; arccos(√3/2) = π/6; arccos(√2/2) = π/4; arccos(1/2) = π/3; arccos(0) = π/2; arccos(−1/2) = 2π/3; arccos(−1) = π. For x = 0.8 (the 4/5 ratio from a 3-4-5 triangle) the result is arccos(0.8) ≈ 0.6435 rad ≈ 36.87°.

As x → ±1 the angle approaches the end of the range and sensitivity (the derivative) grows without bound. Near |x| = 1 a small change in x produces a large change in angle — a geometric feature of the unit circle, not a calculator defect.

Numerical and computational notes

The browser uses arccos (floating-point arithmetic, about 15–16 significant digits). In numerical literature the angle from (x, √(1−x²)) is sometimes computed as atan2(√(1−x²), x), which recovers the same [0, π] branch and can be better conditioned near the endpoints. This calculator reports arccos — the same principal value.

Input outside [−1, 1] is rejected with an error message; the calculator does not return a complex arccos. A comma or a period in the field is accepted equivalently.

Related concepts and authorities

Related tools on this site: inverse sine (range [−π/2, π/2]), inverse tangent, cosine of an angle (the forward map), and the degrees–radians converter. Arccos does not compute triangle side lengths — use the law of cosines for that.

Authorities: NIST DLMF §4.23; Abramowitz & Stegun, Chapter 4; MathWorld — Inverse Cosine; Wikipedia — Inverse trigonometric functions (principal ranges). The calculator is educational and does not replace a textbook.