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

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

question 2

question 3