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.
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| Graph of $y=\csc x$y=cscx |
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| Graph of $y=\sec x$y=secx |
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| 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 $x=\frac{\pi}{2}$x=π2 or at $x=\frac{3\pi}{2}$x=3π2, 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 $x=0,\pi,2\pi,...$x=0,π,2π,... and, to be complete, when $x=\pi n$x=πn, 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$0 | $\frac{\pi}{4}$π4 | $\frac{\pi}{2}$π2 | $\frac{3\pi}{4}$3π4 | $\pi$π | $\frac{5\pi}{4}$5π4 | $\frac{3\pi}{2}$3π2 | $\frac{7\pi}{4}$7π4 | $2\pi$2π |
|---|---|---|---|---|---|---|---|---|---|
| $\cos x$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 $\left[0,2\pi\right]$[0,2π] 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$0 $\frac{\pi}{4}$π4 $\frac{\pi}{2}$π2 $\frac{3\pi}{4}$3π4 $\pi$π $\frac{5\pi}{4}$5π4 $\frac{3\pi}{2}$3π2 $\frac{7\pi}{4}$7π4 $2\pi$2π $\sec x$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 following polynomial, which has a root of multiplicity $3$3.
$P\left(x\right)=x^4-9x^3+30x^2-44x+24$P(x)=x4−9x3+30x2−44x+24.
Find the second derivative of $P\left(x\right)$P(x).
Solve $P''\left(x\right)=0$P′′(x)=0.
Find the root of multiplicity $3$3 of $P\left(x\right)=0$P(x)=0, ensuring working is shown to justify your answer.
Let $x=a$x=a be the other root of $P\left(x\right)=0$P(x)=0. Find the value of $a$a.
Question 3
Consider the graph of $y=\operatorname{cosec}x$y=cosecx below.
When $x=\frac{\pi}{4}$x=π4, $y=\sqrt{2}$y=√2.
What is the next positive $x$x-value for which $y=\sqrt{2}$y=√2?
What is the period of the graph?
What is the smallest value of $x$x greater than $2\pi$2π for which $y=\sqrt{2}$y=√2?
What is the first $x$x-value less than $0$0 for which $y=\sqrt{2}$y=√2?