Intro to sec(x), cosec(x) and cot(x)
The cosecant function at a point $x$x is written as $\csc x$cscx and it is defined by $\csc x=\frac{1}{\sin x}$cscx=1sinx. Similarly, the secant function is defined by $\sec x=\frac{1}{\cos x}$secx=1cosx. And, the cotangent function is defined by $\cot x=\frac{\cos x}{\sin x}$cotx=cosxsinx. The graph of each function is drawn below.
 |
| Graph of $y=\csc x$y=cscx |
 |
| Graph of $y=\sec x$y=secx |
 |
| Graph of $y=\cot x$y=cotx |
All three of these reciprocal trigonometric functions have asymptotes. These occur at points where the relevant parent function $(\sin x$(sinx or $\cos x)$cosx) has value zero. For example, $\sec x=\frac{1}{\cos x}$secx=1cosx is undefined at $x=90^\circ$x=90° or at $x=270^\circ$x=270°, and so on, because at these points $\cos x=0$cosx=0. In addition, all three functions share the same periodicity as their parent functions.
Since $\csc x$cscx and $\sec x$secx are reciprocals of the functions $\sin x$sinx and $\cos x$cosx, the reciprocal functions never attain values strictly between $y=-1$y=−1 and $y=1$y=1. So equations like $\csc x=\frac{1}{2}$cscx=12 have no solutions. This is not true for $\cot x$cotx which can attain any value.
Worked example
At what values of $x$x is the function $y=\cot x$y=cotx undefined?
Think: The function is defined by $\cot x=\frac{\cos x}{\sin x}$cotx=cosxsinx. It is undefined whenever the denominator is zero.
Do: The denominator is zero when $\sin x=0$sinx=0.
This occurs at $x=0^\circ,180^\circ,360^\circ,...$x=0°,180°,360°,... and, to be complete, when $x=180^\circ\times n$x=180°×n, for all integer values of $n$n.
Practice questions
question 1
Consider the identity $\sec x=\frac{1}{\cos x}$secx=1cosx and the table of values below.
| $x$x | $0^\circ$0° | $45^\circ$45° | $90^\circ$90° | $135^\circ$135° | $180^\circ$180° | $225^\circ$225° | $270^\circ$270° | $315^\circ$315° | $360^\circ$360° |
|---|---|---|---|---|---|---|---|---|---|
| $\cos x$cosx | $1$1 | $\frac{1}{\sqrt{2}}$1√2 | $0$0 | $-\frac{1}{\sqrt{2}}$−1√2 | $-1$−1 | $-\frac{1}{\sqrt{2}}$−1√2 | $0$0 | $\frac{1}{\sqrt{2}}$1√2 | $1$1 |
For which values of $x$x in the interval $\left[0^\circ,360^\circ\right]$[0°,360°] is $\sec x$secx not defined?
Write all $x$x-values on the same line separated by commas.
Complete the table of values:
$x$x $0^\circ$0° $45^\circ$45° $90^\circ$90° $135^\circ$135° $180^\circ$180° $225^\circ$225° $270^\circ$270° $315^\circ$315° $360^\circ$360° $\sec x$secx $\editable{}$ $\editable{}$ undefined $\editable{}$ $\editable{}$ $\editable{}$ undefined $\editable{}$ $\editable{}$ What is the minimum positive value of $\sec x$secx?
What is the maximum negative value of $\sec x$secx?
Plot the graph of $y=\sec x$y=secx on the same set of axes as $y=\cos x$y=cosx.
Loading Graph...
question 2
Consider the graphs of $\operatorname{cosec}x$cosecx (black) and $\sec x$secx (grey) below.
In which interval is $\operatorname{cosec}x<0$cosecx<0 and $\sec x>0$secx>0?
$\left(270^\circ,360^\circ\right)$(270°,360°)
A$\left(90^\circ,180^\circ\right)$(90°,180°)
B$\left(0^\circ,90^\circ\right)$(0°,90°)
C$\left(180^\circ,270^\circ\right)$(180°,270°)
D
question 3
Consider the graph of $y=\operatorname{cosec}x$y=cosecx below.
When $x=30^\circ$x=30°, $y=2$y=2.
What is the next positive $x$x-value for which $y=2$y=2?
What is the period of the graph?
What is the smallest value of $x$x greater than $360^\circ$360° for which $y=2$y=2?
What is the first $x$x-value less than $0^\circ$0° for which $y=2$y=2?