Precalculus Review

Tangent function

The function $y = \tan x = \frac{\sin x}{\cos x}$ is called tangent function. The function $\tan x$ is an odd function since $\tan(-x) = \frac{\sin(-x)}{\cos(-x)} = \frac{-\sin x}{\cos x} = -\tan x$ . And it is periodic with period $\pi$ since $\tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin x}{-\cos x} = \tan x$ .

(S) $y = \sin x$

$\includegraphics{lec11a.ps}$

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

$\includegraphics{lec11c.ps}$

The lines $x = \pi/2$ and $x = -\pi/2$ are the vertical asymptotes of $\tan x$ since $\tan x \to \infty$ as $x \to (\pi/2)-$ ( approaches $\pi/2$ from the left), and $\tan x \to -\infty$ as $x \to (-\pi/2)+$ ( approaches $-\pi/2$ from the right). In fact, $x = \dfrac{\pi}{2} + n\pi$ is the vertical asymptote for $\tan x$ for every interger , since it is periodic with period $\pi$ . The domain and the range are given as $D = \left\{x \in\mathbb{R}: x \neq \dfrac{\pi}{2} + n\pi \mbox{ for $n = 0,\pm 1,\pm 2,\ldots$ } \right\}$ and $R = (-\infty, \infty)$ .