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

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The reciprocal functions

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:

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

Each of these is undefined for certain values of $\theta$ . 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.

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

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

{\frac {\sin ^{2}\theta }{\cos ^{2}\theta }}+{\frac {\cos ^{2}\theta }{\cos ^{2}\theta }}={\frac {1}{\cos ^{2}\theta }}

\tan ^{2}\theta +1=\sec ^{2}\theta

There is also a second identity:

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

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

{\frac {\sin ^{2}\theta }{\sin ^{2}\theta }}+{\frac {\cos ^{2}\theta }{\sin ^{2}\theta }}={\frac {1}{\sin ^{2}\theta }}

1+\cot ^{2}\theta =\csc ^{2}\theta

Examples

Question 1:'Find cosec 120, leaving your answer in surd form'

Solution:

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

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

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

Solution:

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

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