The other four trig functions are quotients, so their graphs have a feature sine and cosine never have: vertical asymptotes, where the denominator is zero.
Idea clave
$y = \tan x$ has period $\pi$, zeros at $x = k\pi$, vertical asymptotes at $x = \frac{\pi}{2} + k\pi$, and range all real numbers. $\sec x = \frac{1}{\cos x}$ has asymptotes where $\cos x = 0$; $\csc x$ and $\cot x$ have them where $\sin x = 0$. The period of $A\tan(Bx)$ is $\frac{\pi}{B}$; the period of $A\sec(Bx)$ is $\frac{2\pi}{B}$.
Por qué funciona
$\tan x = \frac{\sin x}{\cos x}$. As $x$ approaches $\frac{\pi}{2}$, $\cos x$ shrinks toward $0$ while $\sin x$ stays near $1$, so the quotient grows without bound: an asymptote. Tangent repeats every $\pi$ (not $2\pi$) because a half turn flips the signs of both sine and cosine, and their quotient stays the same. For secant, the reciprocal of a number between $-1$ and $1$ is at least $1$ in size, so $\sec x$ never lies strictly between $-1$ and $1$.
Worked examples
Ejemplo resuelto: A period
Find the period of $y = 2\tan(3x)$.
- The basic tangent has period $\pi$.
- With $B = 3$, the period is $\frac{\pi}{3}$.
Respuesta: $\frac{\pi}{3}$
Ejemplo resuelto: Asymptotes of secant
Find the vertical asymptotes of $y = \sec x$ on $[0, 2\pi)$.
- $\sec x = \frac{1}{\cos x}$ is undefined where $\cos x = 0$.
- On $[0, 2\pi)$, $\cos x = 0$ at $\frac{\pi}{2}$ and $\frac{3\pi}{2}$.
Respuesta: $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$
Ejemplo resuelto: A squeezed tangent
Find the asymptotes of $y = \tan(2x)$ on $[0, \pi)$.
- $\tan u$ has asymptotes at $u = \frac{\pi}{2} + k\pi$.
- Here $u = 2x$, so $x = \frac{\pi}{4} + \frac{k\pi}{2}$.
- In $[0, \pi)$: $k = 0$ and $k = 1$.
Respuesta: $x = \frac{\pi}{4}$ and $x = \frac{3\pi}{4}$
Error común
Giving tangent the period of sine, $2\pi$. Check two values: $\tan\frac{\pi}{4} = 1$ and $\tan\frac{5\pi}{4} = 1$ as well, only $\pi$ later. The graph repeats every $\pi$.