Cos Inverse Calculator

Enter a cosine value (x) between -1 and 1 to find θ = cos⁻¹(x), the principal angle whose cosine equals x, shown in degrees, radians, and DMS format.

Quick Facts

Formula
θ = cos⁻¹(x), for -1 ≤ x ≤ 1
The principal value always falls between 0° and 180° (0 and π radians).
Key identity
cos(cos⁻¹(x)) = x
Also, cos⁻¹(-x) = 180° − cos⁻¹(x), useful for negative inputs.

Your Results

Calculated
Angle (degrees)
-
θ = cos⁻¹(x), principal value
Angle (radians)
-
θ in radians
Angle (DMS)
-
Degrees, minutes, seconds
Angle classification
-
Acute, right, or obtuse

Ready

Enter a cosine value between -1 and 1, then press Calculate.

Understanding the Inverse Cosine (arccos)

The inverse cosine, written cos⁻¹(x) or arccos(x), answers the question "which angle has a cosine of x?" Given a ratio x between -1 and 1, it returns the single angle θ that satisfies cos(θ) = x. This calculator implements exactly that: enter x, and it solves for θ using the JavaScript Math.acos() function, then converts the result to degrees, radians, and degrees-minutes-seconds.

The formula

For any x with -1 ≤ x ≤ 1:

  • θ = cos⁻¹(x), measured in radians by definition, with 0 ≤ θ ≤ π
  • In degrees: θ = cos⁻¹(x) × 180/π, with 0° ≤ θ ≤ 180°
  • Check: plugging θ back into cosine always returns the original input, cos(cos⁻¹(x)) = x

Because cosine only ever outputs values between -1 and 1, any x outside that interval has no real solution — there is no real angle whose cosine equals 2, for example.

Why the output range is restricted to 0°–180°

Cosine is periodic and even, so many different angles share the same cosine value — cos(60°), cos(-60°), and cos(300°) are all 0.5. A function can only return one output per input, so cos⁻¹ is defined on the restricted, or "principal," range of 0° to 180° (0 to π radians). Within that half-circle, every cosine value from -1 to 1 occurs exactly once, which is what makes the inverse well-defined.

Degrees, radians, and DMS

Radians are the "native" unit for inverse trig functions in mathematics and most programming languages; multiply by 180/π to convert to degrees, or by π/180 to go the other way. Degrees-minutes-seconds (DMS) format — where each degree splits into 60 minutes and each minute into 60 seconds — is still standard in surveying, astronomy, and navigation, so this calculator provides all three side by side.

Common applications

  • Finding an unknown angle in a right triangle when the adjacent side and hypotenuse are known: θ = cos⁻¹(adjacent / hypotenuse)
  • Resolving vector directions in physics and engineering, where a dot-product ratio needs to be converted back to an angle
  • Navigation, surveying, and robotics, where DMS angle output is often required

Frequently Asked Questions

What values can I enter into the cos inverse calculator?
The inverse cosine function cos⁻¹(x) is only defined for x between -1 and 1 inclusive, because cosine itself never produces a value outside that range. Entering a number outside [-1, 1] has no real angle solution and the calculator will flag it as invalid.
Why does cos⁻¹(x) only return angles between 0° and 180°?
Cosine is periodic and repeats the same value for many angles — for example cos(60°) and cos(-60°) are both 0.5 — so it has no single inverse over all angles. To make it a proper function, cos⁻¹ is restricted to the principal range 0° to 180° (0 to π radians), which covers every possible cosine value exactly once.
How do I convert the result from radians to degrees?
Multiply the radian value by 180/π (about 57.2958). This calculator shows both units automatically, along with a degrees-minutes-seconds breakdown used in surveying and navigation.
What is cos⁻¹(0.5) and cos⁻¹(-1)?
cos⁻¹(0.5) equals 60° (π/3 radians), a common reference angle. cos⁻¹(-1) equals 180° (π radians), the maximum possible output since -1 is the smallest valid input.

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