math:functions:tan

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Tangent

Definition

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

Graph

$The graph of the funcrtion $f(x)=\tan(x)$.$

Properties

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:

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