why in matlab sin(pi) is not exact but sin(pi/2) is exact?

Question

I have a problem in calculation with matlab. I know that "pi" is a floating number and is not exact. So, in matlab sin(pi) is not exactly zero. My question is if "pi" is not exact then why sin(pi/2) is exactly equal 1.

sin(pi) --> is not exact beacause of pi. but sin(pi/2) is exactly equal 1

I am wonder and confused!

Answer 1

I dont know the exact way that Matlab calculates sin(x) - but you can investigate this by calculating it using the power series, i.e.

sin x = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + (x^9)/9! ...

Turning this into some Matlab code we represent it by:

clc
x = pi;     %  or x = pi/2
res = x;
factor = -1;
for ii=3:2:19
  res = res + factor*power(x,ii)/factorial(ii);
  factor = factor*-1;  
  fprintf ( 'iteration %2i  sin(x)=%1.16f\n', (ii-1)/2, res );
end
res

Running this code for both x=pi and x=pi/2 you can see that the x=pi/2 converges on the correct result (within eps error) quite quickly (9 iterations) - while the x=pi case doesn't converge in the same time frame.

Its useful to note that at 9 iterations the last factorial that is being calculated in factorial(19). The next factorial that would be calculated in this sequence is 21. This is the last factorial that can be represented with 100% accuracy due to double precision (see help factorial).

So I think whats happening is that for pi/2 the mathematical solution converges on 1 to within double precision quicker that the pi case. In fact the pi case cant converge completely due to limitations in the maths and the accuracy that can be stored in a double precision result.

Having said all that the sin(pi) is within eps so you should use that fact for you purposes.

I've copied the results I get below (R2015b):

Results for PI/2
iteration  1  sin(x)=0.9248322292886504
iteration  2  sin(x)=1.0045248555348174
iteration  3  sin(x)=0.9998431013994987
iteration  4  sin(x)=1.0000035425842861
iteration  5  sin(x)=0.9999999437410510
iteration  6  sin(x)=1.0000000006627803
iteration  7  sin(x)=0.9999999999939768
iteration  8  sin(x)=1.0000000000000437
iteration  9  sin(x)=1.0000000000000000
Final Result: 1.0000000000000000


Results for PI
iteration  1  sin(x)=-2.0261201264601763
iteration  2  sin(x)=0.5240439134171688
iteration  3  sin(x)=-0.0752206159036231
iteration  4  sin(x)=0.0069252707075051
iteration  5  sin(x)=-0.0004451602382092
iteration  6  sin(x)=0.0000211425675584
iteration  7  sin(x)=-0.0000007727858894
iteration  8  sin(x)=0.0000000224195107
iteration  9  sin(x)=-0.0000000005289183
Final Result: -0.0000000005289183