# 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:

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

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

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:

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