About

The inverse cosine function, arccos(x), returns the angle whose cosine equals x. Its domain is restricted to [−1, 1] and its principal value range is [0, π] radians, or [0°, 180°]. Feeding a value outside this domain produces NaN, which silently corrupts downstream calculations in spreadsheets and code. This calculator validates domain membership before computation, returns results in both angle units, and provides a round-trip verification by computing cos(arccos(x)) to confirm numerical consistency.

Note: this tool uses IEEE 754 double-precision arithmetic. Floating-point representation limits precision to roughly 15 - 17 significant digits. Results near domain boundaries (x ≈ ±1) are exact by definition, but intermediate values carry rounding error on the order of 10⁻¹⁵. For safety-critical engineering applications, always cross-check with an independent source.

Formulas

The inverse cosine is defined as the angle θ satisfying:

θ = arccos(x), where −1 ≤ x ≤ 1 and 0 ≤ θ ≤ π

Degree conversion uses the factor:

θ_deg = θ_rad × 180π

The identity linking cosine and its inverse:

cos(arccos(x)) = x, for all x ∈ [−1, 1]

Supplementary angle relationship:

arccos(−x) = π − arccos(x)

Where x = input cosine value, θ = resulting angle, π ≈ 3.14159265.

Reference Data

x	arccos(x) in Degrees	arccos(x) in Radians	Exact Radian Form
−1	180°	3.14159	π
−0.8660	150°	2.61799	5π6
−0.7071	135°	2.35619	3π4
−0.5	120°	2.09440	2π3
0	90°	1.57080	π2
0.2588	75°	1.30900	5π12
0.5	60°	1.04720	π3
0.7071	45°	0.78540	π4
0.8660	30°	0.52360	π6
0.9239	22.5°	0.39270	π8
0.9659	15°	0.26180	π12
1	0°	0	0

Frequently Asked Questions

Cosine is not one-to-one over its full domain, so its inverse must be restricted to a principal branch. The convention uses [0, π] (or [0°, 180°]) because cosine is strictly decreasing and covers its entire range [−1, 1] on that interval. This guarantees a unique output for every valid input.

The real-valued arccos function is undefined outside [−1, 1]. Attempting Math.acos(1.01) in JavaScript returns NaN. This calculator validates the input before computation and displays a domain error. In complex analysis, arccos extends to all real numbers via complex logarithms, but that is outside the scope of this tool.

At domain boundaries, arccos(1) = 0 and arccos(−1) = π are computed exactly because they are special cases in the IEEE 754 implementation. However, values like arccos(0.9999999999999999) may differ from the true analytical result by up to 1 ULP (unit in the last place), approximately 2.2 × 10⁻¹⁶ radians. For most engineering work, this error is negligible.

The general solution is θ = ±arccos(x) + 2nπ, where n is any integer. This calculator returns only the principal value θ₀ ∈ [0, π]. To find all solutions in a specific interval, compute θ₀ and 2π − θ₀, then add integer multiples of 2π as needed.

They are complementary: arccos(x) + arcsin(x) = π/2 for all x ∈ [−1, 1]. This means arccos(0.5) = 60° and arcsin(0.5) = 30°, summing to 90°. The calculator displays both the arccos result and its complement so you can verify this identity.

Yes. The angle between two vectors u and v is θ = arccos((u · v) / (|u| × |v|)). First compute the normalized dot product to obtain x ∈ [−1, 1], then enter that value here. Be cautious with near-parallel vectors: numerical error in the dot product can push x slightly outside [−1, 1], requiring clamping.