 New Zealand
Level 8 - NCEA Level 3

Intro to sec(x), cosec(x) and cot(x)

Lesson

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 $x=\frac{\pi}{2}$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 $\cos x$cosx $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 $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
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.

2. Complete the table of values:

 $x$x $\sec x$secx $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 $\editable{}$ $\editable{}$ undefined $\editable{}$ $\editable{}$ $\editable{}$ undefined $\editable{}$ $\editable{}$
3. What is the minimum positive value of $\sec x$secx?

4. What is the maximum negative value of $\sec x$secx?

5. Plot the graph of $y=\sec x$y=secx on the same set of axes as $y=\cos x$y=cosx.

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)=x49x3+30x244x+24.

1. Find the second derivative of $P\left(x\right)$P(x).

2. Solve $P''\left(x\right)=0$P(x)=0.

3. Find the root of multiplicity $3$3 of $P\left(x\right)=0$P(x)=0, ensuring working is shown to justify your answer.

4. 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.

1. 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?

2. What is the period of the graph?

3. What is the smallest value of $x$x greater than $2\pi$2π for which $y=\sqrt{2}$y=2?

4. What is the first $x$x-value less than $0$0 for which $y=\sqrt{2}$y=2?

Outcomes

M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions