CAIE A-Level Mathematics A2 3.3.1 Reciprocal Trig Functions Questions

Practise equations and graphs involving secant, cosecant and cotangent functions.

Exam points

solve equations involving reciprocal trigonometric functions
use graph intersections to analyse reciprocal trig equations
give solutions in the required interval and units

[Maximum number: 2]

By sketching a suitable pair of graphs, show that the equation $\operatorname{cosec} x=1+\mathrm{e}^{-\frac{1}{2} x}$ has exactly two roots in the interval $0<x<\pi$ .

Sketch a relevant graph, e.g. $y=\operatorname{cosec} x$ $\operatorname{cosec} x$ , U shaped, roughly symmetrical about $x=\frac{\pi}{2}, y\left(\frac{\pi}{2}\right)=1$
and domain at least $\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$ .
Sketch a second relevant graph, e.g. $y=1+\mathrm{e}^{-\frac{1}{2} x}$ , and justify the
given statement
Exponential graph needs y(0)=2, negative gradient, always increasing, and $y(\pi)>1$
Needs to mark intersections with dots, crosses, or say roots at points of intersection, or equivalent

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CAIE A-Level Mathematics A2 3.3.1 Reciprocal Trig Functions Questions question 1