# x = tan(x)

• Nov 6th 2012, 05:55 AM
alyosha2
x = tan(x)
Can some explain to me why xcos(x) - sin(x) will = 0 when x = tan(x)?

Thanks.
• Nov 6th 2012, 05:57 AM
Plato
Re: x = tan(x)
Quote:

Originally Posted by alyosha2
Can some explain to me why xcos(x) - sin(x) will = 0 when x = tan(x)?

$\tan(x)=\frac{\sin(x)}{\cos(x)}$
• Nov 6th 2012, 06:09 AM
alyosha2
Re: x = tan(x)
I understand that, but I don't see how the answer follows from this.
• Nov 6th 2012, 06:24 AM
alyosha2
Re: x = tan(x)
I just don't see it. why is the output of the function needed to be the same as the input? How does that help? What effect does cos or sin have on sinx/cosx? I can't seem to find a pattern. What's the releveance of having tanx * cos(tanx)? Am I thinking about it in the right way? If x and tanx are then interchangeable, what is it that is so special about the value of x at the points where x = tanx that makes it work? I've spend two hours trying to figure this out and I am just completely lost.
• Nov 6th 2012, 06:40 AM
alyosha2
Re: x = tan(x)
I've had an idea. Is it because x and tanx are interchangeable you can use tanx only as a replacement for the first x leaving you with sinx/cosx * cosx - sinx = sinx - sinx = 0?
• Nov 6th 2012, 03:35 PM
Prove It
Re: x = tan(x)
Quote:

Originally Posted by alyosha2
Can some explain to me why xcos(x) - sin(x) will = 0 when x = tan(x)?

Thanks.

\displaystyle \begin{align*} x &= \tan{x} \\ x &= \frac{\sin{x}}{\cos{x}} \\ x\cos{x} &= \sin{x} \\ x\cos{x} - \sin{x} &= 0 \end{align*}
• Nov 6th 2012, 04:57 PM
topsquark
Re: x = tan(x)
Quote:

Originally Posted by Prove It
\displaystyle \begin{align*} x &= \tan{x} \\ x &= \frac{\sin{x}}{\cos{x}} \\ x\cos{x} &= \sin{x} \\ x\cos{x} - \sin{x} &= 0 \end{align*}

Almost. Note that the relation is NOT true when $x = \pi /2$.

@alyosha2 Why does this problem arise with Prove It's proof?

-Dan
• Nov 6th 2012, 05:14 PM
HallsofIvy
Re: x = tan(x)
Quote:

Originally Posted by topsquark
Almost. Note that the relation is NOT true when $x = \pi /2$.

This was all assuming that x= tan(x) which is NOT true when $x= \pi/2$.

Quote:

@alyosha2 Why does this problem arise with Prove It's proof?

-Dan
• Nov 6th 2012, 06:03 PM
Prove It
Re: x = tan(x)
Quote:

Originally Posted by topsquark
Almost. Note that the relation is NOT true when $x = \pi /2$.

@alyosha2 Why does this problem arise with Prove It's proof?

-Dan

I had assumed we were working in the maximal domain implied by the problem: \displaystyle \begin{align*} x \in \mathbf{R} \backslash \frac{n\pi}{2}, n \in \mathbf{Z} \end{align*}.
• Nov 6th 2012, 06:48 PM
topsquark
Re: x = tan(x)
@ Halls and Prove It, good point.

-Dan