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India
Class XI

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 $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}}$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 $\left[0,2\pi\right]$[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 $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 $\editable{}$ $\editable{}$
  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 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.

Loading Graph...

  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

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

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