Arccosine Calculator

This arccosine calculator works out the angle whose cosine equals a value you enter, which is the everyday task of running cosine backwards to recover an angle from a ratio. Arccosine, also written cos⁻¹, is the inverse of the cosine function: if the cosine of an angle is x, arccosine tells you that angle. You enter a single value, which must lie between -1 and 1 inclusive, because cosine itself never produces a result outside that range; enter anything else and the calculator flags that no real angle exists. From that one value it returns the angle in degrees as the main result, the same angle in radians alongside it, and your input value for reference. The answer given is always the principal value, the standard single angle between 0 and 180 degrees (0 and pi radians) that arccosine is defined to return, rather than one of the many other angles that share the same cosine. This makes it useful across trigonometry and geometry problem-solving, physics, surveying and navigation, anywhere you know a ratio and need the angle behind it. Type in a decimal such as 0.5 to see it convert instantly to 60 degrees, and adjust the value to explore how the angle changes as the ratio moves from -1 towards 1. A worked example, the formula and the assumptions behind the calculation are set out below.

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Value (between -1 and 1)

Angle (radians)

Input value

The formula

Arccosine is the inverse of cosine: if cos(theta) = x then arccos(x) = theta. The principal value lies between 0 and 180 degrees (0 and pi radians). The input must be in the range -1 to 1.

Worked example

Arccosine of 0.5 is 60 degrees, because cos 60 = 0.5. Enter 0.5 to confirm.

Frequently asked questions

What does arccosine give me?

The angle whose cosine is the number you enter. It is the inverse of cosine.

What is the valid input range?

The input must be between -1 and 1, because cosine never goes outside that range.

In what range is the angle returned?

Arccosine returns an angle between 0 and 180 degrees.

Who this calculator is for

This calculator is for students, engineers and surveyors who need to find an angle from its cosine.

What this calculator assumes

You enter a value between -1 and 1.
The principal value is returned.
Results are rounded for display.

Formula and sources

Related calculators

Trig Calculator: all six trig functions at once.
Degrees to Radians Calculator: convert angle units.
Right Angle Triangle Calculator: solve a right triangle.

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