sin(2*pi) vs sind(360)

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?

Answer 1

[Mike Croucher]

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.

Docs at Compute cos(X*pi) accurately - MATLAB cospi (mathworks.com) and Compute sin(X*pi) accurately - MATLAB sinpi (mathworks.com)

Answer 2

[Stephen23]

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]

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]

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:

https://en.wikipedia.org/wiki/Floating_point

http://www.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html

http://de.mathworks.com/matlabcentral/answers/57444-faq-why-is-0-3-0-2-0-1-not-equal-to-zero

http://de.mathworks.com/matlabcentral/answers/69-why-does-1-2-3-1-3-not-equal-zero

Answer 3

[Walter Roberson]

Yes, it is due to pi not being represented precisely due to the fact that floating point representation is finite.

[david dang]

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]

For a numerical computation, sin(pi)=1.2246e-16 should be exact enough.

Best wishes

Torsten.