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CIE A-Level Maths Study Notes

2.3.1 Extended Trigonometric Functions

The trigonometry extends beyond sine, cosine, and tangent to include their reciprocals: secant, cosecant, and cotangent. This section explores their definitions, properties, and graphical representations.

Understanding Extended Trigonometric Functions

Secant (sec)

Definition: The secant of an angle in a right-angled triangle is the reciprocal of the cosine, (sec(t)=1cos(t)(\text{sec}(t) = \frac{1}{\cos(t)}.


  • Undefined where cosine is zero (at odd multiples of π2)\frac{\pi}{2} ).
  • Range:
(,1][1,)(-\infty, -1] \cup [1, \infty)
  • An even function: sec(t)=sec(t) \text{sec}(-t) = \text{sec}(t)

Image courtesy of Online Math Learning

Cosecant (csc)

Definition: The cosecant is the reciprocal of sine, csc(t)=1sin(t) \text{csc}(t) = \frac{1}{\sin(t)} .


  • Undefined where sine is zero (at integer multiples of π)\pi ).
  • Shares secant's range.
  • An even function.

Image courtesy of Online Math Learning

Cotangent (cot)

Definition: Cotangent is the reciprocal of tangent,

cot(t)=1tan(t) \text{cot}(t) = \frac{1}{\tan(t)} or cot(t)=cos(t)sin(t)\text{cot}(t) = \frac{\cos(t)}{\sin(t)}


  • Undefined where sine is zero.
  • An odd function:
cot(t)=cot(t)\text{cot}(-t) = -\text{cot}(t)
  • .Range: All real numbers.

Image courtesy of Wikipedia

Application Across Angles of Any Magnitude

Example 1: Understanding Asymptotes

Problem: sec(270)\text{sec}(270^\circ).


Since cos(270)=0,sec(270)\cos(270^\circ) = 0 , \text{sec}(270^\circ) is undefined, indicating a vertical asymptote on the secant graph.

Secant Function

Example 2: Graphical Analysis

Problem: Analyse cot(t)\text{cot}(t) for t t in [0,2π][0, 2\pi].


The cotangent graph shows periodicity and asymptotes at points where sin(t)=0\sin(t) = 0.

Cotangent Function

Example 3: Graph Interpretation

Problem: Interpret the csc(t)\text{csc}(t) graph for tt in [0,2π][0, 2\pi].


The cosecant graph displays inverted U-shaped curves, undefined at t=0t = 0 and t=πt = \pi.

Cosecant Function
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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