Why is √(x^2)/x not equal to sin(x)?

[Albeaver89]

This isn't a homework question, but I felt it was appropriate...

Proof that √(x^2)/x=sin(x)

 

[Mark44]

This isn't a homework question, but I felt it was appropriate...

Proof that √(x^2)/x=sin(x)

You're going to have a very difficult time proving this - it isn't true.

 

[SammyS]

This isn't a homework question, but I felt it was appropriate...

Proof that √(x^2)/x=sin(x)

I think that should be:

[itex]\displaystyle \frac{\sqrt{x^2}}{x}=\text{sign}(x)\ .[/itex]

 

[Albeaver89]

I think that should be:

[itex]\displaystyle \frac{\sqrt{x^2}}{x}=\text{sign}(x)\ .[/itex]

Sorry that's what I thought i had...my bad

 

[lendav_rott]

Oh nevermind, so basically 1 = sinx .
So x = arcsin 1
What is really amasing is how they waste the ink to write X as sqrt(x²)
Unless there's something hidden here, I cannot see the point.
Well sin X = 1 if X is Pi and the way that the sine's sinusoidal graph repeats itself you can get the other possibilities for X.

 

[SammyS]

Sorry that's what I thought i had...my bad

Edit your original post, & make a note there that you've edited it.

 

[Mark44]

Oh nevermind, so basically 1 = sinx .

No for a couple of reasons. First, the OP meant sign(x) not sin(x). Second,
## \frac{\sqrt{x^2}}{x} \neq 1##

So x = arcsin 1
What is really amasing is how they waste the ink to write X as sqrt(x²)

No. x ≠ ##\sqrt{x^2}##

Unless there's something hidden here, I cannot see the point.
Well sin X = 1 if X is Pi and the way that the sine's sinusoidal graph repeats itself you can get the other possibilities for X.

 

[lendav_rott]

Wait, x =/= sqrt(x²)??

I cannot see what you are trying to say - do you mean that sqrt(x²) = +/- X?
This is actually something I was arguing over with my math's lector and he said that the square root of X or X² for that matter, is defined as sqrt(X²) = |X|

And after thinking about it, i thought about

A^x = B
x lnA = lnB
if B were negative then this wouldn't hold true and the only explanation is that Sqrt(A²) = |A|

 

[Mark44]

Wait, x =/= sqrt(x²)??

Yes, that's exactly what I mean. As an example, do you think that ##\sqrt{(-2)^2} = -2##?

I cannot see what you are trying to say - do you mean that sqrt(x²) = +/- X?

No, I don't mean that either. The square root of a nonnegative expression produces a single value, not two of them, as ± x implies.

This is actually something I was arguing over with my math's lector and he said that the square root of X or X² for that matter, is defined as sqrt(X²) = |X|

Your lecturer is correct.

And after thinking about it, i thought about

A^x = B
x lnA = lnB
if B were negative then this wouldn't hold true and the only explanation is that Sqrt(A²) = |A|

Also, your second step isn't valid if A ≤ 0, because ln(A) wouldn't be defined.

 

[Akshay_Anti]

the reason why x is not equal to root of x squared is this-

If a, b are positive numbers, and you take root of that (in complex numbers, of course), then the relation √-a × √-b = √(-×-)ab = √(+ab) does not hold valid

instead, √-a × √-b = i√a × i√b = -√ab

that's the reason... and also, √x² / x should be equal to signum function of x which equals modulus of x divided by x