Arccos Calculator

What this calculator does

So, you know the cosine of an angle and you want the angle itself. This tool reverses cosine: you enter a cosine value, and it returns the angle, in radians and in degrees, to three decimal places.

One input box holds the cosine value. This is the cosine calculator working backwards, taking a ratio and handing back the angle it came from.

How to use it

1. Type the cosine value, a number between -1 and 1.
2. Press Calculate.

The result appears in both radians and degrees together, so you can read off whichever unit suits your problem.

Arccosine undoes cosine

Cosine turns an angle into a ratio. Arccosine, written arccos or cos with a small -1, turns that ratio back into an angle. The name reads as "the arc, or angle, whose cosine is this number." So arccos(0.5) means "the angle whose cosine is 0.5", which is 60 degrees. It is the tool you reach for whenever a cosine value is known, often from the ratio of two sides or from a formula, and the angle is what you actually want.

The input range and the output range

Two ranges matter here, and they are different from each other. The input, like arcsine, must sit between -1 and 1, because cosine, being a coordinate on a circle of radius 1, never goes beyond those bounds. Ask for the arccosine of 2 and the tool refuses, since no angle has such a cosine.

The output range, though, is not the same as arcsine's. Arccosine returns a principal angle between 0 and 180 degrees, a half-turn's worth, rather than arcsine's -90 to 90. This is the standard choice that lets arccosine give exactly one angle for every cosine value from -1 to 1, sweeping from 0 degrees where the cosine is 1, through 90 degrees where it is 0, to 180 degrees where it is -1.

The link to arcsine: they add to 90 degrees

Arccosine and arcsine are tied together by a neat rule. For any value between -1 and 1, the arcsine and the arccosine of that value add up to exactly 90 degrees. So if the arcsine of 0.5 is 30 degrees, the arccosine of 0.5 must be 60 degrees, and indeed it is. This mirrors the complement relationship between sine and cosine themselves, and it means you can always get one from the other by subtracting from 90 degrees.

A worked example

Enter 0.5. The tool returns the angle whose cosine is 0.5, which is 60 degrees, or about 1.047 radians. Enter 1 and it returns 0 degrees, since the cosine is 1 only when the angle is 0. Enter -1 and it returns 180 degrees, the far end of the range. Enter 0 and it returns 90 degrees.

Radians and degrees

Both units come back at once. Degrees are the familiar everyday measure; radians measure the angle by arc length on the unit circle and are standard in higher maths. For arccosine, the degree answer always lands between 0 and 180, and the radian answer between 0 and pi.

Questions people ask

What does arccosine do?

It takes a cosine value and returns the angle that has that cosine. It is the inverse of the cosine function.

Why must the input be between -1 and 1?

Because cosine never leaves that range, so no angle has a cosine outside it. An input beyond those bounds has no matching angle.

Why is the output range 0 to 180, not -90 to 90?

That is the standard principal range for arccosine, chosen so that every cosine value from -1 to 1 maps to exactly one angle. It differs from arcsine's range on purpose.

How is it related to arcsine?

For any value, its arcsine and arccosine add to 90 degrees. So you can find one by subtracting the other from 90.

Which unit is the answer in?

Both radians and degrees, shown together, so you can use whichever you need.

References

On the inverse cosine and its principal range. Arccosine is the inverse of the cosine, defined for inputs from -1 to 1, returning a single angle between 0 and 180 degrees.

1. Eric W. Weisstein, "Inverse Cosine," from MathWorld, a Wolfram resource, on the inverse function of the cosine and its principal values.
2. Eric W. Weisstein, "Inverse Trigonometric Functions," from MathWorld, a Wolfram resource, on the inverse trigonometric functions and the conventions for their ranges.