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Inverse Cos Calculator | Compute arccos for inputs -1 to 1

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Convert a cosine ratio to its angle with arccos, in degrees or radians, including the supplementary solution. Shows the principal range and the unit-circle reasoning — ideal for right-triangle trigonometry and physics problems.

Last updated: July 2026 · Free · No sign-up required

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Results

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What the Inverse Cos Calculator does

Every result on this page comes from a real formula — θ = cos⁻¹(x), defined for −1 ≤ x ≤ 1; result ranges from 0° to 180° — computed live in your browser the moment you press the button.

For context, the sections beneath the calculator include typical values, a worked example you can recompute by hand, and an FAQ covering the practical details of inverse cosine calculator.

Quick reference

xarccos(x)
10°
0.86630°
0.70745°
0.560°
090°
−0.5120°
−1180°

info Inverse Cos Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

OutputMeaning
DegreesThe angle whose cosine equals x, between 0° and 180°.
RadiansThe same angle in radians (0 to π).
Domainx must be between −1 and 1; outside that, arccos is undefined.

lightbulb Worked example — the angle of a 10 ft ladder

A 10 ft ladder stands with its base 6 ft from the wall. The angle θ between the ladder and the ground has cos θ = adjacent/hypotenuse = 6/10 = 0.6.

Result: θ = cos⁻¹(0.6) = 53.13°. In radians: 53.13 × π ÷ 180 = 0.927 rad. Sanity check: cos(53.13°) = 0.6, confirming the angle.

What this means: 53° is a fairly shallow ladder angle — safe ladder guidance usually targets around 75° from the ground (a 4:1 height-to-base ratio), so this ladder should be moved closer to the wall before climbing. The arccos result tells you the actual set-up angle from a simple base measurement.

Formula & methodology

Formula: θ = cos⁻¹(x), defined for −1 ≤ x ≤ 1; result ranges from 0° to 180°

How the inverse cosine is computed

Arccos answers “what angle has this cosine?” It is the inverse of cosine on the principal range 0°–180° (0 to π radians), where each cosine value corresponds to exactly one angle.

  • The input x must lie between −1 and 1 — outside that range, arccos is undefined for real angles.
  • The calculator evaluates θ = cos⁻¹(x) in radians, the native output of the math library.
  • It converts to degrees with θ × 180 ÷ π and reports both.
  • The supplementary angle 180° − θ covers the second solution when a full-circle answer is needed.

Example: cos⁻¹(0.5) = 60°, since cos 60° = 0.5. cos⁻¹(−0.5) = 120°, the supplementary angle.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is the inverse cosine of a number?
The Inverse Cos Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Calculates inverse cosine using θ = cos⁻¹(x), defined for −1 ≤ x ≤ 1, live in your browser. There is no limit on usage, and it works on any device.
What is the domain of arccos?
The Inverse Cos Calculator is a free online tool that calculates inverse cosine using θ = cos⁻¹(x), defined for −1 ≤ x ≤ 1, live in your browser. You enter a few values, and it returns an accurate result together with a step-by-step explanation, so it works as both a calculator and a quick reference for the underlying concept.
What range does arccos return?
Arccos returns angles between 0° and 180° (0 to π radians), the principal range chosen so the function is single-valued. A cosine of 1 gives 0°, 0 gives 90°, and −1 gives 180°. This differs from arcsin, which returns −90° to 90°. The calculator shows the principal angle and, where relevant, the supplementary solution.
Is cos⁻¹ the same as 1/cos?
No — cos⁻¹(x) is the inverse cosine function, which answers 'what angle has this cosine?', while 1/cos(x) is the secant, the reciprocal of the cosine VALUE. The notation is confusingly similar but the meanings are completely different: cos⁻¹(0.5) = 60°, while 1/cos(60°) = 2. The calculator applies the inverse function, not the reciprocal.
Is it free and private to use?
Yes — the Inverse Cos Calculator. There is no sign-up, no paywall, no trial, and no limit on how many calculations you can run. CalcaTools covers its costs with non-intrusive display advertising, so the calculator itself never asks for payment or restricts any feature.
What do cos⁻¹, arccos, and acos mean — are they different?
They're three notations for the same inverse cosine function. cos⁻¹(0.5) = 60° whichever way it's written. The trap is reading cos⁻¹ as 1/cos — that reciprocal is the secant, a different function entirely. On most calculators you reach it with SHIFT/2nd + COS.
Why does inverse cosine say 'math error' for my number?
Arccos only accepts inputs from −1 to 1, because cosine never leaves that range. If you're solving a triangle and get cos θ = 1.15, the triangle as specified is impossible — usually a sides mix-up in the law of cosines. Also check the mode: arccos(0.5) is 60 in degrees but 1.047 in radians.
How is arccos used with the law of cosines?
The law of cosines gives you cos C = (a² + b² − c²) ÷ 2ab from three side lengths, and arccos converts that ratio into the actual angle. For sides 5, 6, 7: cos C = (25 + 36 − 49) ÷ 60 = 0.2, so C = cos⁻¹(0.2) = 78.46°. It is the standard way to recover angles from measured sides.
What is the difference between arccos and the secant?
Arccos (cos⁻¹) asks ‘which angle has this cosine?’ and returns an angle; the secant (sec x = 1/cos x) takes an angle and returns a ratio. The ⁻1 superscript means inverse function, not reciprocal — cos⁻¹(0.5) = 60°, while sec(60°) = 2. Confusing the two is the most common notation trap in trigonometry.
Can arccos handle negative inputs?
Yes — any input from −1 to 1 is valid, and negative inputs give obtuse angles between 90° and 180°. cos⁻¹(−0.5) = 120°, because cos 120° = −0.5. The ‘math error’ only appears for inputs outside [−1, 1], which usually means a triangle’s sides were entered inconsistently.
When would I want the answer in radians instead of degrees?
In calculus and physics, where angle formulas assume radians — the derivative of sin x is cos x only in radians, and arc-length s = rθ needs θ in radians. Degrees are for human-scale geometry (construction, navigation); radians are for the mathematics that follows. Both are reported so you can take whichever your next step needs.

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