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India
Class X

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 $x=90^\circ$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}}$12 $0$0 $-\frac{1}{\sqrt{2}}$12 $-1$1 $-\frac{1}{\sqrt{2}}$12 $0$0 $\frac{1}{\sqrt{2}}$12 $1$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.

  2. 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 $\editable{}$ $\editable{}$
  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.

    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?

Loading Graph...

  1. $\left(270^\circ,360^\circ\right)$(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.

Loading Graph...
The function $y=\csc x$y=cscx is plotted on a Cartesian plane with x- and y-axes labeled. The x-axis ranges from 0 to 360, marked in major intervals of 90 and minor intervals of $30$30. The y-axis ranges from -2 to 2, marked in major intervals of 1 and minor intervals of $\frac{1}{3}$13. A vertical dashed line is drawn at x=0 and a horizontal dashed line at y=0.
  1. When $x=30^\circ$x=30°, $y=2$y=2.

    What is the next positive $x$x-value for which $y=2$y=2?

  2. What is the period of the graph?

  3. What is the smallest value of $x$x greater than $360^\circ$360° for which $y=2$y=2?

  4. What is the first $x$x-value less than $0^\circ$0° for which $y=2$y=2?

Outcomes

10.T.IT.1

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.

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