sin(2*pi) vs sind(360)

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david dang on 13 Jul 2015
Answered: Mike Croucher on 21 Oct 2022
Could someone please explain to me why sin(2*pi) gives me non-zero number and sind(360)? Does this have to do with the floating points of pi?

Mike Croucher on 21 Oct 2022
If you ever need to compute sin(x*pi) or cos(x*pi), its better to do sinpi(x) or cospi(x). You never explicitly multily x by a floating point approximation ot pi so you always get the results you expect.

Stephen23 on 13 Jul 2015
Edited: Stephen23 on 13 Jul 2015
Yes, it is because π is a value that cannot be represented precisely using a finite binary floating point number. This is also shown in the sind documentation:
"Sine of 180 degrees compared to sine of π radians"
sind(180)
ans =
0
sin(pi)
ans =
1.2246e-16
david dang on 13 Jul 2015
Thanks for the answer. I tried setting pi = sym(pi), but this increased my computation time significantly. Is there any way to perform my computations in radians, without increasing computation time significantly, and obtain the exact solution?
Stephen23 on 13 Jul 2015
Edited: Stephen23 on 13 Jul 2015
No.
Unless of course you buy a computer with infinite memory to hold an infinite representation of π and yet can somehow perform operations at the same speed as your current computer.
π is an irrational number. How do you imagine representing an irrational number with a finite floating point value and not getting rounding error? All numeric computations with floating point numbers include rounding errors, and it is your job to figure out how to take this into account. To understand floating point numbers you should read these:

Walter Roberson on 13 Jul 2015
Yes, it is due to pi not being represented precisely due to the fact that floating point representation is finite.
david dang on 13 Jul 2015
Thanks for the answer. I tried setting pi = sym(pi), but this increased my computation time significantly. Is there any way to perform my computations in radians, without increasing computation time significantly, and obtain the exact solution?
Torsten on 13 Jul 2015
For a numerical computation, sin(pi)=1.2246e-16 should be exact enough.
Best wishes
Torsten.