Tangent, cotangent, secant, cosecant

Each of the following is a ratio or reciprocal of sine or cosine functions, and so is not defined at values of $t$ when the denominator is zero. The result is that the graphs of these functions have vertical asymptotes at those singular ("bad") values, shown in red in the graphs below.

$\displaystyle \frac{(t)}{(t)}$ #[The tangent of $t$ is defined by][La tangente de $t$ de define por]# \t

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

\\ \t

#[Graph of][Gráfica de]# $y = \tan(t)$

\\ $\displaystyle \frac{(t)}{(t)}$ #[The cotangent of $t$ is defined by][la cotangente de $t$ de define por]# \t

$\displaystyle \cotan(t) = \frac{\cos(t)}{\sin(t)}$

\\ \t

#[Graph of][Gráfica de]# $y = \cotan(t)$

\\ $\displaystyle \frac{1}{(t)}$ #[The secant of $t$ is defined by][la secante de $t$ de define por]# \t

$\displaystyle \sec(t) = \frac{1}{\cos(t)}$

\\ \t

#[Graph of][Gráfica de]# $y = \sec(t)$

\\ $\displaystyle \frac{1}{(t)}$ #[The cosecant of $t$ is defined by][la cosecante de $t$ de define por]# \t

$\displaystyle \cosec(t) = \frac{1}{\sin(t)}$

\\ \t

#[Graph of][Gráfica de]# $y = \cosec(t)$

Examples

1. $\displaystyle \tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0$, whereas \\ 2. $\displaystyle \tan\left(\frac{\pi}{2}\right) = \frac{\sin(\pi/2)}{\cos(\pi/2)}$ #[is not defined, as][no es definida, porque]# $\displaystyle \cos\left(\frac{\pi}{2}\right)=0$. \t #[Notice that the graph of $\tan$ has a vertical asymptote at][Observe que la gráfica de $\tan$ tiene una asíntota vertical en]# $\dfrac{\pi}{2}.$ \\ 3. $\displaystyle \tan\left(\frac{\pi}{2}\right) = \frac{\sin(\pi/2)}{\cos(\pi/2)}$ #[is not defined, as][no es definida, porque]# $\displaystyle \cos\left(\frac{\pi}{2}\right)=0.$ \\ 4. $\displaystyle \tan\left(\frac{\pi}{4}\right) = \cotan\left(\frac{\pi}{4}\right) = 1,$ #[as][porque]# $\displaystyle \sin\left(\frac{\pi}{4}\right)=cos\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$. \\ 5. $\displaystyle \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos(\pi/3)} = \frac{1}{1/2} = 2$. \\ 6. $\displaystyle \cosec(0) = \frac{1}{\sin(0)}$ #[is not defined, as][no es definida, porque]# $\sin(0)=0,$ #[whereas][aunque]# \\ 7. $\displaystyle \sec(0) = \frac{1}{\cos(0)} = \frac{1}{1} = 1$

#[Some for you][Algunas para ti]#

#[Evaluate the following. ][Evalua las siguentes. ]#

◼$\color{purple}{y=\sin x}$ ◼$\color{teal}{y=\cos x}$

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