Solve for general solutions
$\tan(x/3) = 1$
When I solve this equation my answer comes to be $x = 3\pi/4 \pm 2n\pi, 15\pi/4 \pm 2n\pi$ where $n$ is an integer
However when I graph the equation $y = \tan(x/3) - 1$ values for $x$ such as $11\pi/4$ do not equal zero.
What would $x$ be equal to then?
Hint:
General solution of $\tan x$ is
$$\tan { x } =a\\ x=\arctan { a } +k\pi ,k\in \mathbb{Z} $$
where is period is $\pi $ not $2\pi $
The general solution can be described as $$\frac x3\equiv \frac \pi 4\pmod \pi\iff x\equiv \frac{3\pi}4\pmod{3\pi}$$ since the tangent function has period $\pi$.
$\tan\left(\dfrac{x}{3}\right)=1$
$\dfrac{x}{3}=\dfrac{\pi}{4}+n\pi,\;\forall n\in\mathbb{Z}$
and this becomes
$x=\dfrac{3}{4}\pi+3n\pi,\;\forall n\in\mathbb{Z}$
This explains why solutions are $\dfrac{3}{4}\pi,\;\dfrac{15 \pi }{4},\dfrac{27 \pi }{4},\ldots$ etc
