Value of Sine(π) - Calc&Graph on Google Play

[Adit]

I know it should be absolute zero, but in a scientific calculator, which is an app actually, its value is shown 1.224646799x10^(-16). The value is so small, but it is not zero. I wanted to be sure that this is a bug or really it has a value. App's name is Calc&Graph, on Google play.

 

[maajdl]

You could say it is a bug.
You could also consider that any calculator only returns an approximate value.
Often, on handlheld calculators, the approximation for trig functions is based on the CORDIC algorithm.
http://en.wikipedia.org/wiki/CORDIC
It is much more precise than a MacLaurin development would be, because it relies on exact trig relations.
However, the arithmetic calculations are carried out with a finite number of digits (bits) and therefore even multiplications or divisions are approximate.

You need also to take care of the precision of the data you have input to the calculator.
For example, this is an "exact" result:

Sin[3.14159265358979] = 3.23109*10^-15

but it could be that the precision of the calculator can't handle more decimals after 3.1415...879 .

 

[pasmith]

I know it should be absolute zero, but in a scientific calculator, which is an app actually, its value is shown 1.224646799x10^(-16). The value is so small, but it is not zero. I wanted to be sure that this is a bug or really it has a value. App's name is Calc&Graph, on Google play.

$\sin(\pi) = 0$.

Calculators do not understand irrational numbers; they can only ever deal with a finite subset of the rational numbers. Occasionally this will cause them to get things wrong. But a decent calculator should be able to tell the difference between "sine of (user pressed the pi button)" and "sine of (some number starting with 3.14)" and will return the correct value for the former.

This can also cause discrepancies between "user wants the cube root of -27, apparently that's -3" and "user wants to raise -27 to the power 0.333333333333; I don't know how to do that with a negative base".

 

[Adit]

Oh thank you, I got it. That is calculating value of sine(3.14...), and the close it is to π, closer my answer is to zero. Thanks, I didn't think of that.

 

[blue_raver22]

Thank you for bringing this to my attention. The value of sine(π) should indeed be zero, as you correctly stated. However, the value of 1.224646799x10^(-16) is not a bug, but rather a result of the limitations of computer and calculator precision. This value is known as machine epsilon, which is the smallest number that can be represented on a computer. Essentially, the calculator is rounding the value to the nearest number it can accurately represent, which in this case is 1.224646799x10^(-16). So while technically not zero, this value is as close to zero as the calculator can show. It is important to keep in mind that all calculations done on a computer or calculator are subject to these limitations of precision.