Arccos Calculator

Cosine value (x)

Angle

deg (°)

Angle in radians (multiples of pi)
Sine of the angle (sqrt of 1 minus x squared)
Complementary angle (arcsin of x): deg (°)

The cosine of 60 deg (°) is 0.5, which confirms the inverse: arccos undoes cosine.

Arccos always returns the principal angle between 0 and 180 degrees (0 to pi radians), even though many angles share the same cosine.

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Show the arccos graph

Plot the inverse cosine curve from x = -1 to x = 1.

Show the angle in every unit

List your angle in degrees, radians, multiples of pi, gradians, and turns.

Your angle in every unit
Unit	Your angle
Degrees	60 deg
Radians	1.047198 rad
Radians (multiples of pi)	0.3333 pi
Gradians	66.6667 grad
Turns	0.166667 turn

The arccos calculator tool determines the angle whose cosine is equal to a given value. The cosine takes an angle and gives a ratio from -1 to 1. Arccos works in the opposite direction, it takes a value from -1 to 1 and gives the corresponding angle. You can either enter a cosine value to retrieve its corresponding angle or enter an angle to obtain its cosine. The units for angles (degrees, radians, gradians, revolutions) can be changed at any time.

What is Arccos?

Arccos is written as arccos(x) and can also be interpreted as the inverse operation of cosine on x. It is the inverse function of the cosine function. An inverse function undoes a previous function. If the cosine of an angle is x, then the arccosine of x is that angle.

\cos(\theta) = x \quad\Longleftrightarrow\quad \arccos(x) = \theta

There are two important restrictions to note: the input values must be in the range of -1 to 1. This is because cosine cannot produce ratios outside this range. Also, since cosine repeats every 360 degrees, many angles will have the same cosine value. To provide a unique value, arccos returns the principal value, which is an angle between 0 and 180 (or 0 and pi radians).

Note the notation as a negative exponent of cosine. This represents an inverse function and not a reciprocal. The reciprocal of cosine (1 over cosine) is secant, and they are two different things.

Here's how to use this calculator.

In the first input field you can enter a cosine in the range of -1 to 1 and the corresponding angle will be shown immediately. The result is displayed as a heading and also in the field for the angle. There you can switch between degrees, radians, gradians and revolutions. The panel also shows the angle as a multiple of pi, the sine of this angle and the complementary angle.

This tool allows for two-way calculations. If you leave the cosine field blank and enter an angle into the Angle field, it will calculate the cosine of that angle. By opening the graph you can see the entire arccos curve, and by opening the units table you can read off angles in different units at the same time.

Inverse cosine formula:

This relationship can be expressed in a single line of code. The angle is the arccosine of x, and the cosine of that angle returns x.

\theta = \arccos(x) \qquad \cos(\theta) = x

Let's do an example with the calculator. The cosine of 60 degrees is equal to 0.5, so the arccosine of 0.5 is equal to 60 degrees, which is equivalent to radians when you divide pi by 3.

\cos(60^\circ) = 0.5 \quad\Longrightarrow\quad \arccos(0.5) = 60^\circ = \frac{\pi}{3}

Now for the arccos of 0.5

Calculating angles in a right-angled triangle.

When two sides are known, the arccos function is a common method for calculating angles in a right triangle. The cosine of an angle equals the division of the adjacent side by the longest side (hypotenuse), so this angle will be equal to the arccos of that ratio.

\beta = \arccos\!\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)

Suppose the length of the side adjacent to the angle is 52 and the hypotenuse is 60. The ratio is found by dividing 52 by 60 which gives approximately 0.866. So this angle is equal to arccos(0.866), or 30 degrees.

\cos(\beta) = \frac{52}{60} \approx 0.866 \quad\Longrightarrow\quad \beta = \arccos(0.866) = 30^\circ

If the length of the adjacent side is not known yet, it must be calculated first using the Pythagorean Theorem and then divided by the hypotenuse before taking the arccos.

Common values for arccos:

There are some numbers that come up again and again. This table shows these numbers both in degrees and radians. If you enter a cosine value, the calculator may show one of these values.

Cosine value (x)	Angle in degrees	Angle in radians
1	0°	$0$
0.866	30°	$\frac{\pi}{6}$
0.707	45°	$\frac{\pi}{4}$
0.5	60°	$\frac{\pi}{3}$
0	90°	$\frac{\pi}{2}$
-0.5	120°	$\frac{2\pi}{3}$
-0.707	135°	$\frac{3\pi}{4}$
-0.866	150°	$\frac{5\pi}{6}$
-1	180°	$\pi$

Useful identities for arccos.

With some trigonometric identities it is possible to simplify the calculation of arccos and also understand its relationship with other inverse trigonometric functions.

\cos(\arccos(x)) = x

\sin(\arccos(x)) = \sqrt{1 - x^2}

\arcsin(x) + \arccos(x) = \frac{\pi}{2}

The last identity shows that arccos and arcsin are complementary angles. The sum of the angle x in cosine and the angle x in sine is always 90 degrees. There are also addition formulas, which allow you to combine two values of arccos into one.

\arccos(\alpha) + \arccos(\beta) = \arccos\!\left(\alpha\beta - \sqrt{(1 - \alpha^2)(1 - \beta^2)}\right)

Applications of arccos

Arccos converts ratios to angles. This is a process that's required in many fields. In geometry it's used along with the law of cosines to determine the angles of a triangle when all three sides are known. In physics and computer graphics, it's used to calculate the angle between two vectors from their dot product, which forms a basis for lighting, camera calculations, and collision detection. It's also used in signal processing to extract phase angles from normalized amplitudes.

Hints and precautions

acos only accepts inputs between -1 and 1. If the calculated ratio is slightly outside of this range due to rounding errors, e.g. becomes 1.0000001, you need to bring it back into range before calculating acos.
The result is always the principal value and lies in the range of 0 to 180 degrees. To find an angle in another quadrant with the same cosine, subtract the principal value from 360 degrees.
Degrees are used in everyday life and engineering while radians are used in analysis. The unit conversion function allows you to convert between the two units.
There are three values worth remembering: The arccosine of 1 is 0 degrees, the arccosine of 0 is 90 degrees and the arccosine of -1 is 180 degrees.
The negative powers of cosine are the inverse function, not the result of dividing 1 by cosine. The reciprocal of cosine is secant.

Frequently asked questions

What is the domain of arccos?

The domain is between -1 and 1. Since the cosine does not produce values outside of this range, the inverse function cannot accept values outside of this range either. For results like 1.0000001 that are produced by rounding floating point numbers, there will be no real solutions; so correct the values first to bring them within the allowable range.

Is cosine with a negative exponent equal to the result of dividing the cosine by 1?

No, it is not the same thing. This notation can be confusing. A negative exponent on cos represents the inverse function arccos. The reciprocal of cosine, or 1 divided by cosine, is secant. These are different functions that happen to use the same symbols, which can lead to confusion.

What is the relationship between arccos and arcsin?

They are complementary angles to each other. The arcsine of x is equal to 90 degrees minus the arccosine of x; thus their sum always equals 90 degrees or pi radians divided by 2. Arccosine gives the angle whose cosine is x, while arcsine gives the angle whose sine is x.

What is the arccosine of 0?

90 degrees, which is the radian measure that you get when you divide pi by 2. And since cosine of 90 degrees is equal to 0, going in the opposite direction, arccosine of 0 is equal to 90 degrees. This is exactly halfway through our output range from 0 to 180 degrees.

Is it possible to determine arccosine without a calculation tool?

Yes, if it's a commonly used value then you can write the equation "cosine of angle equals x" and recall or look up the angle whose cosine is equal to x. The arccosine of 0.5 is 60 degrees, the arccosine of 0 is 90 degrees. For more complex inputs, trigonometric tables or a calculator are practical solutions.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Inverse trigonometric functions
Definitions, principal values, domains, and identities for arccos and its siblings.
Math is Fun: Inverse Sine, Cosine, Tangent
Plain-language explanation of how the inverse trig functions undo sine, cosine, and tangent.