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

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The reciprocal functions[edit]

The graph of y=csc(x) in radians
The graph of y=sec(x) in radians
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:

Each of these is undefined for certain values of . 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.

Dividing both sides by

There is also a second identity:

Dividing both sides by


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


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


  • If
  • If