What is the Relationship Between Sine and Inverse Pi as n Increases?

[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.

 

[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 to see how to derive the result.

 

[dimension10]

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 to see how to derive the result.

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

 

[micromass]

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

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!