Reciprocal Trig Functions - Graphs (AQA A-Level Mathematics): Revision Notes

5.5.2 Reciprocal Trig Functions - Graphs

The graphs of the reciprocal trigonometric functions—cosecant $\csc \theta$, secant $\sec \theta$, and cotangent $\cot \theta$—are derived from the basic trigonometric functions sine, cosine, and tangent, respectively. Understanding these graphs is important for visualising the behaviour of these functions and recognising their key features such as asymptotes and periodicity.

1. Graph of Cosecant ($\csc \theta$):

• Definition: $\csc \theta = \frac{1}{\sin \theta}$
  • The cosecant function is undefined where $\ \sin \theta = 0$, which occurs at $\theta = 0^\circ, 180^\circ, 360^\circ, \dots .$
• Asymptotes:
  • Vertical asymptotes occur where $\sin \theta = 0 ,$ so the graph has vertical asymptotes at $\theta = 0^\circ, 180^\circ, 360^\circ, \dots .$
• Shape:
  • Between these asymptotes, the graph has branches that mirror the sine wave but are flipped and stretched.
  • The graph approaches infinity as $\theta$ approaches the asymptotes from either side.
  • At the peaks of the sine function (where ($\sin \theta = 1 \ or \ -1$), the cosecant function will intersect at $y = 1$ or ($y = -1$), respectively.
• Periodicity:
  • The period of the $\csc \theta$ function is the same as $\ \sin \theta$, which is $\ 360^\circ$ or $2\pi$ radians.

2. Graph of Secant ($\sec \theta$):

• Definition: $\sec \theta = \frac{1}{\cos \theta}$
  • The secant function is undefined where $\cos \theta = 0$, which occurs at $\theta = 90^\circ, 270^\circ, 450^\circ, \dots .$
• Asymptotes:
  • Vertical asymptotes occur where $\cos \theta = 0$, so the graph has vertical asymptotes at $\ \theta = 90^\circ, 270^\circ, 450^\circ, \dots .$
• Shape:
  • The graph of $\ \sec \theta$ consists of upward and downward branches that mirror the cosine wave, flipped and stretched.
  • At the points where$\ \cos \theta = \pm 1$ (the maximum and minimum values of cosine), the secant function intersects at $\ y = 1$ and $\ y = -1$, respectively.
  • As $\theta$ approaches the asymptotes, the secant function approaches infinity.
• Periodicity:
  • The period of the $\sec \theta$ function is the same as $\cos \theta$, which is $\ 360^\circ \ or \ 2\pi$ radians.

3. Graph of Cotangent ($\cot \theta$):

• Definition: $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$
  • The cotangent function is undefined where $\tan \theta = 0$, which occurs at $\theta = 0^\circ, 180^\circ, 360^\circ, \dots .$
• Asymptotes:
  • Vertical asymptotes occur where $\sin \theta = 0$, so the graph has vertical asymptotes at $\theta = 0^\circ, 180^\circ, 360^\circ, \dots .$
• Shape:
  • The graph of $\cot \theta$ is a decreasing curve between each pair of vertical asymptotes, reflecting the fact that the cotangent function decreases as $\theta$ increases within each interval.
  • Unlike the tangent function, which increases from negative to positive infinity, the cotangent function decreases from positive to negative infinity as $\theta$ increases.
• Periodicity:
  • The period of the $\cot \theta$ function is $180^\circ$ or $\pi$ radians, which is half the period of the sine and cosine functions.

Summary of Reciprocal Trig Function Graphs: