Reciprocal Trigonometric Functions (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

Use the definitions of the secant, cosecant and cotangent functions to show that

$\text{sec}\theta \text{cot}\theta \equiv \text{cosec}\theta$

Hence solve the equation

$\text{sec}\theta \text{cot}\theta =-2$

for $0\le \theta \le 2\pi$.

Show that the equation

$3-\text{sec}\theta =\frac{2}{\text{sec}\theta }$

can be expressed in the form

$\left(\text{sec}\theta -2\right)\left(\text{sec}\theta -1\right)=0$

Hence solve the equation

$3-\text{sec}\theta =\frac{2}{\text{sec}\theta }$

for $0\le \theta \le 2\pi$.

Using the double angle formula $\text{sin}2A\equiv 2 \text{sin}A \text{cos}A$, show that the equation

$\text{sec}x\text{cosec}x-5=\text{cosec}2x$

can be expressed in the form

$\text{cosec}2x=5$

Hence solve the equation

$\text{sec}x\text{cosec}x-5=\text{cosec}2x$

for $0\le x\le 2\pi$, giving your answers correct to 3 significant figures.

Show that the equation

${\text{tan}}^{2}x=6\text{sec}x-10$

can be expressed in the form

${\left(\text{sec}x-3\right)}^2=0$

Hence solve the equation

${\text{tan}}^{2}x=6\text{sec}x-10$

for $0\le x\le 2\pi$, giving your answers correct to 3 significant figures.

(i) Sketch, in the interval $-2\pi \le \theta \le 2\pi$, the graph of $y=3+2\text{cosec}\theta$, including asymptotes and the coordinates of all maximum and minimum points.

(ii) Hence state the number of solutions of the equation $3+2\text{cosec}\theta =\frac{1}{2}$ in this interval.

Express $\text{tan}\theta \text{cosec}\theta$ as a single trigonometric function.

Hence solve the equation

$\text{tan}\theta \text{cosec}\theta =-\frac{2\sqrt{3}}{3}$

for $-\pi <\theta \le \pi$, giving your answers as exact values.

Show that the equation

$2{\text{cot}}^{2}x=1-5\text{cosec}x$

can be expressed in the form

$\left(2\text{cosec}x-1\right)\left(\text{cosec}x+3\right)=0$

Hence solve the equation

$2{\text{cot}}^{2}x=1-5\text{cosec}x$

for $0\le x\le 2\pi$, giving your answers correct to 3 significant figures.

Solve the equation

$\frac{2}{\text{cosec}\theta }-\text{cosec}\theta =1$

for $0\le \theta \le 2\pi$.

Using the double angle formula $\text{sin}2A\equiv 2 \text{sin}A \text{cos}A$, solve the equation

$\text{sec}x\text{cosec}x-75=5\text{cosec}2x$

for $-\pi <x\le \pi$, giving your answers correct to 3 significant figures.

(i) Sketch, in the interval $-2\pi \le \theta \le 2\pi$, the graph of $y=-5+\frac{1}{2}\text{sec}\theta$, including asymptotes and the coordinates of all maximum and minimum points.

(ii) Hence state the set of values of $k$ for which the equation $-5+\frac{1}{2}\text{sec}\theta =k$ has no solutions.

Solve the equation

$\frac{\text{sec}\theta \text{cot}\theta }{\text{cosec}\theta \text{tan}\theta }=-\sqrt{3}$

for $-\pi <\theta \le \pi$.

Solve the equation

$6\text{sec}\theta +\frac{2\sqrt{3}}{\text{sec}\theta }=-3-4\sqrt{3}$

for $0\le \theta \le 2\pi$, giving your answers in an exact form.

Solve the equation

$3{\text{cot}}^{2}x-4\sqrt{3}=\left(6-2\sqrt{3}\right)\text{cosec}x-3$

for $0\le x\le 2\pi$, giving your answers in an exact form.

Using the double angle formulae $\text{sin}2A\equiv 2\text{sin}A\text{cos}A$ and $\text{cos}2A\equiv {\text{cos}}^{2}A-{\text{sin}}^{2}A$, solve the equation

$\left(\text{cosec}x-\text{sec}x\right)\left(\frac{1}{\text{sec}x}+\frac{1}{\text{cosec}x}\right)=\text{cot}2x+3$

for $-\pi <x\le \pi$, giving your answers correct to 3 significant figures.

Sketch, in the interval $-2\pi \le \theta \le \pi$, the graph of $y=2+3\text{sec}\left(\theta +\frac{\pi }{2}\right)$, including asymptotes and the coordinates of all maximum and minimum points.

Deduce the maximum and minimum values of $\frac{1}{2+3\text{sec}\left(\theta +{\displaystyle \frac{\pi }{2}}\right)}$.