Reciprocal Trigonometric Functions (Cambridge International AS Mathematics): Exam Questions

Sketch the graph of $y = \text{cosec} x$, for $- 180 ^{\circ} \leq x \leq 180 ^{\circ}$.

Solve the equation $\text{cot} x = 3$ for $- π \leq x \leq π$, giving your answers correct to 3 significant figures.

Sketch the graph of $y = \text{sec} θ$, for $- π \leq θ \leq π$.

Label any points of intersection with the coordinate axes and state the equations of any asymptotes.

Starting with the identity

$\text{sin}^{2} x + \text{cos}^{2} x \equiv 1$

show that

(i) $1 + \text{cot}^{2} x \equiv \text{cosec}^{2} x$

(ii) $\text{tan}^{2} x + 1 \equiv \text{sec}^{2} x$

Show that

$\text{sec}^{2} θ \text{sin} θ \equiv \text{tan} θ \text{sec} θ$

Solve the equation

$\text{cosec}^{2} x - 2 \text{cosec} x - 8 = 0$

for $0 ^{\circ} \leq x \leq 360 ^{\circ}$, giving your answers correct to 1 decimal place where appropriate.

Show that

$\text{cot} x \text{cosec} x \text{sec} x \equiv 1 + \text{cot}^{2} x$

Solve the equation

$\text{sec} θ \text{tan} θ - \text{sec} θ = 0$

for $0 \leq θ \leq 2 π$, giving your answers in an exact form.

Sketch the graph of $y = 2 \text{sec} 2 x$ for $- π \leq x \leq π$.

By sketching a suitable straight line on your diagram, show that the equation $2 \text{sec} 2 x = 4$ has four solutions in this interval.

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

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

Hence solve the equation

$\text{sec} θ \text{cot} θ = - 2$

for $0 \leq θ \leq 2 π$.

Show that the equation

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

can be expressed in the form

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

Hence solve the equation

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

for $0 \leq θ \leq 2 π$.

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

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

can be expressed in the form

$\text{cosec} 2 x = 5$

Hence solve the equation

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

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

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

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

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

Hence solve the equation

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

for $- π < θ \leq π$, 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 $0 \leq x \leq 2 π$, giving your answers correct to 3 significant figures.

Solve the equation

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

for $0 \leq θ \leq 2 π$.

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

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

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

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

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

Solve the equation

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

for $- π < θ \leq π$.

Solve the equation

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

for $0 \leq θ \leq 2 π$, 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 $0 \leq x \leq 2 π$, giving your answers in an exact form.

Using the double angle formulae $\text{sin} 2 A \equiv 2 \text{sin} A \text{cos} A$ and $\text{cos} 2 A \equiv \text{cos}^{2} A - \text{sin}^{2} A$, 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} \; 2 x + 3

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

Sketch, in the interval $- 2 π \leq θ \leq π$, 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 + \frac{\pi}{2}\right)}.