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

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The reciprocal functions[edit]

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[edit]

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[edit]

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 = 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