IB Maths AA 3.8 Notes

The reciprocal trigonometric functions are defined by:

• $\sec x=\frac{1}{\cos x}$
• $\cosec x=\frac{1}{\sin x}$
• $\cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x}$

These are useful because they appear naturally when rearranging identities and solving equations.

Since they are reciprocals:

• $\sec x$ is undefined when $\cos x=0$
• $\cosec x$ is undefined when $\sin x=0$
• $\cot x$ is undefined when $\tan x=0$, equivalently when $\sin x=0$

We can see this below:

Further to this, reciprocal functions have a useful graphical relationship with $\sin x$, $\cos x$ and $\tan x$.

For a reciprocal function:

• it equals $1$ where the original function equals $1$
• it equals $-1$ where the original function equals $-1$
• it becomes very large positive or negative where the original function approaches $0$
• zeros of the original function become vertical asymptotes of the reciprocal graph

Thus, with the reciprocal trigonometric functions:

• $y=\sec x$ has vertical asymptotes where $\cos x=0$
• $y=\cosec x$ has vertical asymptotes where $\sin x=0$
• $y=\cot x$ has vertical asymptotes where $\tan x=0$