Question about significant figures.

[Woozie]

This isn't really a homework question. It's a question about a book I'm reading out of curiosity, but this seemed to be the most appropriate place to put it.

Also, it's not exactly a physics problem, but I did come across this issue while reading a physics related book.

I apologize if this is the wrong section to post this question.

But my question is: How do you keep track of significant figures for functions in general? For example, how would I know how many significant figures to keep in Sin(3.52)? Would I automatically keep three significant figures as with multiplication? Or is there a different rule for this? How would I do this for functions in general that are not directly addition/subtraction or multiplication/division?

I just realized that none of my physics or math books explains this. I also realized that all these years, I've been taking trig functions and other types of functions for granted when calculating significant figures.

 

[bob1182006]

Well if you can expand a trig function to it's series representation.

for sin x you have:
$$\sin x = \sum_{n=1}^\infty (-1)^{n-1} \frac{x^{2n-1}}{(2n-1)!}$$

the first few terms for sin 3.52:

n=1,2,3

$$3.52-\frac{(3.52)^3}{3!}+\frac{(3.52)^5}{5!}$$

=3.52-(-3.75)+4.50=.754

which is pretty far from the real value (.061396...)

But from the multiplication/additions you will be keeping 3 sig figs, since in the addition there's a large number of digits after the decimal point. But the multiplication (3.52^2=3.52*3.52) limits the number down to 3 sig figs here.

 

[mgb_phys]

I would treat them as a multiplication - 3s.f. of angle gives 3 s.f. of sin.
In reality is complicated because it is a function of angle, in regions where the sin is changing quickly (0,180) the uncertainty of the angle has a much bigger effect on the sin than regions near 90,270 deg .