Tangent function

The function

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

is called a tangent function. The function $ \tan x$ is an odd function, since

$\displaystyle \tan(-x) = \dfrac{\sin(-x)}{\cos(-x)}
= \dfrac{-\sin x}{\cos x} = -\tan x
$

(S) $ y = \sin x$

\includegraphics{lec11a.ps}

(C) $ y = \cos x$

\includegraphics{lec11b.ps}

(T) $ y = \tan x = \dfrac{\sin x}{\cos x}$

\includegraphics{lec11c.ps}

Period. The tangent function is periodic with period $ \pi$, since

$\displaystyle \tan(x + \pi) = \dfrac{\sin(x + \pi)}{\cos(x + \pi)}
= \dfrac{-\sin x}{-\cos x} = \tan x.
$

Vertical asymptotes. The lines $ x = \pi/2$ and $ x = -\pi/2$ are the vertical asymptotes of $ \tan x$, since

$\displaystyle \tan x \to \infty$    as $ x \to (\pi/2)-$    and $\displaystyle \quad
\tan x \to -\infty$    as $ x \to (-\pi/2)+$.

Recall that `` $ x \to (\pi/2)-$'' means ``$x$ approaches $ \pi/2$ from the left,'' and `` $ x \to (-\pi/2)+$'' means ``$x$ approaches $ -\pi/2$ from the right.'' Then

$\displaystyle x = \dfrac{\pi}{2} + n\pi$    for every interger $n$

is the vertical asymptote for $ \tan x$, since it is periodic with period $ \pi$.

Domain and range. $ D = \left\{x \in\mathbb{R}: x \neq \dfrac{\pi}{2} + n\pi
\mbox{ for $n = 0,\pm 1,\pm 2,\ldots$ } \right\}$ and $ R = (-\infty, \infty)$.



Department of Mathematics
Last modified: 2005-09-29