Why does math.acos result in ValueError: math domain error?

Question

I am using math.acos() function :

math.acos(1.0000000000000002)

This throws a math domain error. Can someone tell the reason? I am getting this value calculated before and here this value gives error but if I remove 2 at the end it does not throw error. I did not get the reason to this.

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The cosine of an angle is always in the range `[-1.0, 1.0]`, so the inverse function is only defined for inputs in that range. You're giving `acos` a value larger than `1`: there's no possible (real) angle whose cosine is greater than `1`. — Mark Dickinson, Jun 20 '15 at 14:29
The cosine of an angle in a right triangle is the relation between the length of the closest leg and the hypotenuse. Since the hypotenuse is always greater than any of the legs, the cosine of an angle is always less than one. Arc-cos, or "acos", is the inverse function. You're telling it you have a triangle where the hypotenuse is shorter than the closest leg, and now asking python what angle that would give. Python throws a fit, since what you are asking is impossible! :-) — avl_sweden, Nov 30 '19 at 15:18
I encountered this issue when calculating `math.acos(numpy.dot(a,b))` where `numpy.dot` returned `1.0000000000000002` instead of 1 (due to floating point precision). See [this post](https://stackoverflow.com/questions/20457038/how-to-round-to-2-decimals-with-python) for solutions to this. — DaBooba, May 21 '20 at 21:21

score 15 · Accepted Answer · edited Jan 07 '23 at 08:17

15

You are trying to do acos of a number for which the acos does not exist.

Acos - Arccosine , which is the inverse of cosine function.

The value of input for acos range from -1 <= x <= 1 .

Hence, when trying to do math.acos(1.0000000000000002), you are getting the error.

If you try higher numbers, you will keep getting the same error - math.acos(2) leads to ValueError: math domain error

edited Jan 07 '23 at 08:17

wjandrea

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answered Jun 20 '15 at 14:30

Anand S Kumar

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score 3 · Answer 2 · answered Jun 20 '15 at 14:31

Inverse cosine is defined only between -1 and 1, inclusive. The arc-cosine of 1.0000000000000002 has no mathematical or semantic meaning other than "does not exist" or "undefined".

Of course, since inverse cosine of 1 does exist, acos(1) doesn't throw any error.

Why does math.acos result in ValueError: math domain error?

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