Question on Double Angle theroem

[grapz]

Solve for X in domain ( 0, 2 pi)

4 sin x cos x - 1 = 0

so i factor out a 2

2 ( 2 sin x cos x ) -1 = 0

2 ( sin 2x ) - 1 = 0

sin 2x = 1/2

now this is the part where i do not understand.

x is restricted to domain ( 0, 2 pi)

but , it tells me that 2 x is restricted to the domain (0 , 4 pi ).

Can some one explain to me why the domain is doubled? I mean i can see that its 2x , so its doubled, but i dont' get the logic behind it.

thanks

 

[daon]

If $\displaystyle sin(2x) = 1/2$ then either $\displaystyle 2x = \pi/6$ or $\displaystyle 2x=5\pi/6$. If you consider the next possible correct answer, $\displaystyle 2x=2\pi+\pi/6=13\pi/6$ then $\displaystyle x=13\pi/12$, and is still in the interval \(\displaystyle \[0, \2\pi]\). Same for $\displaystyle 2x=17\pi/6$.

That is what the domain for 2x is "twice the size" (although these words are technically incorrect to use) than that of x.