The Arccos Calculator computes the inverse cosine (arccosine) of any value in the domain [−1, 1], returning the principal value angle in degrees and radians. Used across trigonometry, geometry, physics, and engineering wherever an angle must be recovered from its cosine value.
60
°
1.047198
rad
60
°
1.047198
rad
You know the cosine of an angle. You need the angle itself. That is exactly what the inverse cosine function does — and the calculator for arccos computes arccos(x) for any input in the valid domain [−1, 1], returning the principal value angle in both degrees and radians with full precision.
The arccosine function is the inverse of the cosine function, restricted to the principal value range:
arccos(x) = θ such that cos(θ) = x and 0° ≤ θ ≤ 180°
The domain is −1 ≤ x ≤ 1 (since cosine values are bounded between −1 and +1). Key reference values:
Inputs outside [−1, 1] are undefined (not a real number) — no angle has a cosine greater than 1 or less than −1. Use this online calculator for any value in the valid range. The arcsin calculator computes the inverse sine for the complementary function.
The Law of Cosines uses arccos to find angles in any triangle from its side lengths:
cos(C) = (a² + b² − c²) / (2ab) → C = arccos[(a² + b² − c²) / (2ab)]
For a triangle with sides a = 5, b = 7, c = 8: cos(C) = (25 + 49 − 64) / (2 × 5 × 7) = 10/70 = 0.1429; C = arccos(0.1429) = 81.79°. The Law of Cosines generalizes the Pythagorean theorem — when C = 90°, cos(C) = 0, and a² + b² − c² = 0, recovering the familiar c² = a² + b². The unit circle calculator provides the complete trigonometric values for all standard angles.
Arccos appears wherever an angle must be recovered from a cosine relationship:
The reference angle calculator and trigonometry calculators cover the complete inverse trig and angle function toolkit.
arccos(x) is not the same as 1/cos(x) = sec(x). The notation cos⁻¹(x) unambiguously means the inverse function (arccos), not the reciprocal. This distinction matters: arccos(0.5) = 60°, while 1/cos(60°) = 1/0.5 = 2 — completely different operations. The inverse trig functions (arccos, arcsin, arctan) undo the trigonometric operation; the reciprocal trig functions (sec, csc, cot) are ratios. When you see cos⁻¹ on a calculator button, it always means arccos, not secant.
Enter a value between -1 and 1. The calculator applies the inverse cosine function to compute the angle whose cosine equals your input value, displaying results in both degrees and radians.
The result is the unique angle in the range $$[0°, 180°]$$ whose cosine equals your input value. A result near 0° means the input is close to 1, while a result near 180° means the input is close to -1. An input of 0 always yields exactly 90°.
Inputs
Results
Since cos(60°) = 0.5, the arccosine of 0.5 is exactly 60° or π/3 radians.
Inputs
Results
Since cos(150°) = −√3/2 ≈ −0.866, the result is 150° or 5π/6 radians.
They are identical. Both $$\arccos(x)$$ and $$\cos^{-1}(x)$$ denote the inverse cosine function. Be careful not to confuse $$\cos^{-1}(x)$$ with $$\frac{1}{\cos(x)}$$, which is the secant function $$\sec(x)$$.
The cosine function decreases monotonically from 1 to -1 on the interval $$[0°, 180°]$$. This interval is chosen as the principal range because it ensures a one-to-one mapping, making the inverse function well-defined and unique for every input.
They are complementary: $$\arccos(x) + \arcsin(x) = 90°$$ (or $$\frac{\pi}{2}$$ radians) for every valid input $$x \in [-1,1]$$. So $$\arccos(x) = 90° - \arcsin(x)$$.
$$\arccos(0) = 90°$$ or $$\frac{\pi}{2}$$ radians, because $$\cos(90°) = 0$$.
Given vectors $$\mathbf{a}$$ and $$\mathbf{b}$$, the angle between them is $$\theta = \arccos\left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right)$$, where the numerator is the dot product and the denominator is the product of their magnitudes.
$$\int \arccos(x)\,dx = x\,\arccos(x) - \sqrt{1-x^2} + C$$. This result is obtained using integration by parts.
How helpful was this calculator?
5.0/5 (1 rating)
