# Sine of inverse pi pattern

1. Jun 21, 2011

### dimension10

I have noticed something strange when you take the value of sin(pi*10^-n). It approaches pi*10^-n. I have attatched the file here.

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• ###### Sine theorem of inverse powers of pi.pdf
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2. Jun 21, 2011

### micromass

Hi dimension10!

Your result can be generalized. Indeed, if x is small then

$\sin(x)\sim x$

So for small values of x, we will have that x approximates sin(x) quite closely.

The precise result is

$\lim_{x\rightarrow 0}{\frac{\sin(x)}{x}}=1$

which can be proved by geometric methods. See http://www.khanacademy.org/video/proof--lim--sin-x--x?playlist=Calculus [Broken] to see how to derive the result.

Last edited by a moderator: May 5, 2017
3. Jun 21, 2011

### dimension10

So that just means that sin(0)/0=1, right?

Last edited by a moderator: May 5, 2017
4. Jun 21, 2011

### micromass

No, not at all, since you cannot divide by 0. What

$$\lim_{x\rightarrow 0}{\frac{\sin(x)}{x}}=1$$

mean is, if x is very close to 0 (but not equal to 0!!), then $\frac{\sin(x)}{x}$ comes very close to 1.
Thus if x is very close to 0, then sin(x) comes very close to x!!

The statement sin(0)/0 makes no sense, since division by 0 is not allowed!