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.
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 $$n.
Consider the identity $$secx=1cosx and the table of values below.
$$x | $$0° | $$45° | $$90° | $$135° | $$180° | $$225° | $$270° | $$315° | $$360° |
---|---|---|---|---|---|---|---|---|---|
$$cosx | $$1 | $$1√2 | $$0 | $$−1√2 | $$−1 | $$−1√2 | $$0 | $$1√2 | $$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.
Complete the table of values:
$$x | $$0° | $$45° | $$90° | $$135° | $$180° | $$225° | $$270° | $$315° | $$360° |
---|---|---|---|---|---|---|---|---|---|
$$secx | $$ | $$ | undefined | $$ | $$ | $$ | undefined | $$ | $$ |
What is the minimum positive value of $$secx?
What is the maximum negative value of $$secx?
Plot the graph of $$y=secx on the same set of axes as $$y=cosx.
Consider the graphs of $$cosecx (black) and $$secx (grey) below.
In which interval is $$cosecx<0 and $$secx>0?
$$(270°,360°)
$$(90°,180°)
$$(0°,90°)
$$(180°,270°)
Consider the graph of $$y=cosecx below.
When $$x=30°, $$y=2.
What is the next positive $$x-value for which $$y=2?
What is the period of the graph?
What is the smallest value of $$x greater than $$360° for which $$y=2?
What is the first $$x-value less than $$0° for which $$y=2?