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India
Class X

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

Lesson

The cosecant function at a point $$x is written as $$cscx and it is defined by $$cscx=1sinx. Similarly, the secant function is defined by $$secx=1cosx. And, the cotangent function is defined by $$cotx=cosxsinx. The graph of each function is drawn below.

Graph of $$y=cscx

 

Graph of $$y=secx

 

Graph of $$y=cotx

 

All three of these reciprocal trigonometric functions have asymptotes. These occur at points where the relevant parent function $$(sinx or $$cosx) has value zero. For example, $$secx=1cosx is undefined at $$x=90° or at $$x=270°, and so on, because at these points $$cosx=0. In addition, all three functions share the same periodicity as their parent functions.

Since $$cscx and $$secx are reciprocals of the functions $$sinx and $$cosx, the reciprocal functions never attain values strictly between $$y=1 and $$y=1. So equations like $$cscx=12 have no solutions. This is not true for $$cotx which can attain any value.

Worked example

At what values of $$x is the function $$y=cotx undefined?

Think: The function is defined by $$cotx=cosxsinx. It is undefined whenever the denominator is zero.

Do: The denominator is zero when $$sinx=0.

This occurs at $$x=0°,180°,360°,... and, to be complete, when $$x=180°×n, for all integer values of $$n.

Practice questions

question 1

Consider the identity $$secx=1cosx and the table of values below.

$$x $$0° $$45° $$90° $$135° $$180° $$225° $$270° $$315° $$360°
$$cosx $$1 $$12 $$0 $$12 $$1 $$12 $$0 $$12 $$1
  1. For which values of $$x in the interval $$[0°,360°] is $$secx not defined?

    Write all $$x-values on the same line separated by commas.

  2. Complete the table of values:

    $$x $$0° $$45° $$90° $$135° $$180° $$225° $$270° $$315° $$360°
    $$secx $$ $$ undefined $$ $$ $$ undefined $$ $$
  3. What is the minimum positive value of $$secx?

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

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

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question 2

Consider the graphs of $$cosecx (black) and $$secx (grey) below.

In which interval is $$cosecx<0 and $$secx>0?

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  1. $$(270°,360°)

    A

    $$(90°,180°)

    B

    $$(0°,90°)

    C

    $$(180°,270°)

    D

question 3

Consider the graph of $$y=cosecx below.

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

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

  2. What is the period of the graph?

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

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

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