$$f(x)=\tan(x)\equiv \frac{ \sin(x) } { \cos(x) }.$$

• The function $\tan$ is periodic with period $\pi$, not $2\pi$ like $\sin$ and $\cos$.
• The $\tan$ function has asymptotes at all values of $x$ for which the denominator ($\cos$) goes to zero.
  • The locations of the asymptotes are $x=\frac{\pi}{2},\frac{-\pi}{2},\frac{\pi}{2},\frac{3\pi}{2},\ldots$.
  • At those values, $\tan$ approaches $\infty$ from the left, and $-\infty$ from the right.
• Value at $0$: $\tan(0)=\frac{0}{1}=0$ because $\sin(0)=0$.
• The angle $x=\frac{\pi}{4}$ is special since both $\sin$ and $\cos$ are equal

and we get:

  \[
    \tan\left(\frac{\pi}{4} \right) 
     = \frac{ \sin\left(\frac{\pi}{4}\right) }{ \cos\left(\frac{\pi}{4}\right) }
     = \frac{ \frac{\sqrt{2}}{2}  }{ \frac{\sqrt{2}}{2}  }
     = 1.
  \]