# A-level Mathematics/MEI/C4/Trigonometry/Reciprocal trigonometrical functions

< A-level Mathematics‎ | MEI‎ | C4‎ | Trigonometry

## The reciprocal functions

File:Y=csc(x).gif
The graph of y=csc(x) in radians
File:Y=sec(x).gif
The graph of y=sec(x) in radians
File:Y=cot(x).gif
The graph of y=cot(x) in radians

Aside from the classic 3 trigonmetical functions, there are now 3 more you must be aware of; the reciprocals of our standard ones. We have the cosecant (csc), secant (sec), and cotangent (cot). These are defined as:

• ${\displaystyle \csc \theta ={\frac {1}{\sin \theta }}}$
• ${\displaystyle \sec \theta ={\frac {1}{\cos \theta }}}$
• ${\displaystyle \cot \theta ={\frac {1}{\tan \theta }}={\frac {\cos \theta }{\sin \theta }}}$

Each of these is undefined for certain values of ${\displaystyle \theta }$. For example; cscθ is undefined when θ=0,180,360..., because sinθ=0 at these points.

Each of the graphs of these functions all have asymptotes intervals of 180 degrees.

## Some new identities

Using our new definitions of reciprocal functions, we are able to obtain 2 new identities based of Pythagoras' theorem.

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$

Dividing both sides by ${\displaystyle \cos ^{2}\theta }$

${\displaystyle {\frac {\sin ^{2}\theta }{\cos ^{2}\theta }}+{\frac {\cos ^{2}\theta }{\cos ^{2}\theta }}={\frac {1}{\cos ^{2}\theta }}}$
${\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }$

There is also a second identity:

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$

Dividing both sides by ${\displaystyle \sin ^{2}\theta }$

${\displaystyle {\frac {\sin ^{2}\theta }{\sin ^{2}\theta }}+{\frac {\cos ^{2}\theta }{\sin ^{2}\theta }}={\frac {1}{\sin ^{2}\theta }}}$
${\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }$

## Examples

Solution:

• ${\displaystyle \csc 120={\frac {1}{sin120}}}$
• ${\displaystyle =1/{\frac {\sqrt {3}}{2}}}$
• ${\displaystyle ={\frac {2}{\sqrt {3}}}}$

Question 2:'Find all values of x in the interval 0≤x≤360 for:

${\displaystyle \sec ^{2}x=4+2\tan x}$

Solution:

• ${\displaystyle \sec ^{2}x=4+2\tan x}$
• ${\displaystyle \tan ^{2}x+1=4+2\tan x}$
• ${\displaystyle \tan ^{2}x-2\tan x-3=0}$
• ${\displaystyle (\tan x-3)(\tan x+1)=0}$
• ${\displaystyle \tan x=3or\tan x=-1}$
• If ${\displaystyle \tan x=3}$
• ${\displaystyle x=71.6,251.6}$
• If ${\displaystyle \tan x=-1}$
• ${\displaystyle x=135,315}$
• ${\displaystyle x=71.6,135,251.6,315}$