Question

Find all solutions to the equation in radians:

2(sqrt3) - 4 = -4 - 2tan(theta/3)

Answer 1

I can rewrite your equation as below:
\[2\sqrt{3}=-2 \tan(\theta/3)\]
so:
\[\tan(\theta/3)=-\sqrt{3}\]
whose solution are:

\[\frac{ \theta }{ 3 }=\frac{ 5 \pi }{ 3 }+k*\pi\]
so:
\[\theta=5 \pi+3*k*\pi\]
where:
\[k=0, \pm1, \pm2, \pm3....\]

Answer 2

I'm confused... where did the 5pi/3 come from? The unit circle?

Answer 3

yes!

you can choose:
\[\frac{ \theta }{ 3 }=\frac{ 2 \pi }{ 3 }+k*\pi\]
or
\[\frac{ \theta }{ 3 }=\frac{ 5 \pi }{ 3 }+k*\pi\]
as I did!