Further Trigonometric Equations (Cambridge International AS Mathematics): Exam Questions

Solve the equation $\text{sec} θ = 1$ for $0 ^{\circ} \leq θ \leq 360 ^{\circ}$.

Solve the equation

\text{tan} \left( A - 30° \right) = \tfrac{1}{3}\sqrt{3}

for $- 180 ^{\circ} \leq A \leq 180 ^{\circ}$.

Solve the equation

$\frac{1}{\text{sec}x} = \frac{1}{2} \sqrt{2}$

for $- π \leq x \leq π$, giving your answers in terms of $π$.

Express

$8 \text{cos} θ + 6 \text{sin} θ$

in the form R \; \text{cos} \left( \theta - \alpha \right), where $R > 0$ and $0 < α < \frac{1}{2} π$.

State the value of $R$ and give the value of $α$ correct to 3 significant figures.

Show that the equation

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

can be expressed in the form

\left( \text{cosec} \; x - 1 \right)^2 = 0

Hence solve the equation

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

for $- 2 π \leq x \leq 2 π$, giving your answers in terms of $π$.

Solve the equation

$\text{cos} 2 θ = \frac{1}{2}$

for $- π \leq θ \leq π$, giving your answers in terms of $π$.

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

Use your graph to find the set of values of $k$ for which the equation

$\text{sec} x = k$

has no real roots for $- π \leq x \leq π$.

Solve the equation

$\text{cot}^{2} θ - \text{cos} θ \text{cosec}^{2} θ = 0$

for $0 < θ < 2 π$, giving your answers in terms of $π$.

Solve the equation $\text{cosec} 2 θ = 2$ for $0 ^{\circ} \leq θ \leq 180 ^{\circ}$.

Solve the equation

\text{tan} \left( 3A - 30° \right) = -\tfrac{1}{3}\sqrt{3}

for $- 120 ^{\circ} \leq A \leq 120 ^{\circ}$.

Solve the equation

$\frac{\text{sin}x}{\text{sec}x} = \frac{1}{4}$

for $- π \leq x \leq π$, giving your answers in terms of $π$.

Use the identity

R \; \text{sin} \left( A + B \right) \equiv R \; \text{cos} \; B \; \text{sin} \; A + R \; \text{sin} \; B \; \text{cos} \; A

to show that

$3 \text{sin} θ + 4 \text{cos} θ$

can be expressed in the form

5 \; \text{sin} \left( \theta + \alpha \right)

where

\alpha = \text{tan}^{-1} \left( \tfrac{4}{3} \right)

Hence solve the equation

$3 \text{sin} θ + 4 \text{cos} θ = 1$

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

State the greatest value of $3 \text{sin} θ + 4 \text{cos} θ$ and find the least positive value of $θ$ for which this greatest value occurs. Give your value of $θ$ correct to 3 significant figures.

Show that the equation

$3 \text{tan}^{2} x = 18 - 2 \text{sec} x$

can be expressed as

$3 \text{sec}^{2} x + 2 \text{sec} x - 21 = 0$

Hence solve the equation

$3 \text{tan}^{2} x = 18 - 2 \text{sec} x$

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

Solve the equation

$\text{cos} 2 θ = \text{cos} θ - 1$

for $- π \leq θ \leq π$, giving your answers in terms of $π$.

Sketch the graph of $y = \text{cot}^{2} θ$ for $- 2 π \leq θ \leq 2 π$.

By drawing suitable horizontal lines on your sketch, show how the number of real roots of the equation

$\text{cot}^{2} θ = k$

for $- 2 π \leq θ \leq 2 π$, where $k$ is a constant, can be $0$, $4$ or $8$.

Solve the equation

$\text{cot}^{2} θ = \text{sec}^{2} θ - 1$

for $0 ^{\circ} \leq θ \leq 360 ^{\circ}$.

Solve the equation $\text{sec}^{2} 2 x = 1 + \text{tan} 2 x$ for $0 ^{\circ} \leq x \leq 180 ^{\circ}$.

Given that

\text{sin} \left( 2A - B \right) = \tfrac{1}{4} \left( \sqrt{6} + \sqrt{2} \right)

where $3 A = 4 B$ and $60 ^{\circ} < B < A < 300 ^{\circ}$, find the value of $A$ and the value of $B$.

Solve the equation

$\frac{\text{cos}x}{\text{cosec}x} - \text{cot} x = 0$

for $- 2 π \leq x \leq 2 π$, giving your answers in terms of $π$.

Show that $6 \text{cos} θ - 8 \text{sin} θ$ can be expressed in the form R \; \text{cos} \left( \theta + \alpha \right), where $R > 0$ and $α$ is an acute angle measured in radians.

Hence solve the equation

$3 \text{cos} θ - 4 \text{sin} θ - 2 = 0$

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

State the least value of $6 \text{cos} θ - 8 \text{sin} θ$ and find the least positive value of $θ$ for which this least value occurs. Give your value of $θ$ correct to 3 significant figures.

Solve the equation

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

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

Solve the equation

$8 \text{cos}^{4} θ - 5 \text{cos} 2 θ - 2 = 0$

for $0 \leq θ \leq π$, giving your answers in terms of $π$.

Find the set of values of the constant $k$ for which the equation

$\text{cosec} θ = k$

for $- π \leq θ \leq 2 π$, has

(i) no real roots,

(ii) exactly 1 real root,

(iii) exactly 2 real roots,

(iv) exactly 4 real roots.

Solve the equation

$\text{cot}^{2} θ = 15 - 6 \text{cosec} θ$

for $- 180 ^{\circ} \leq θ \leq 180 ^{\circ}$, giving your answers correct to 1 decimal place where appropriate.

Solve the equation

$\text{sin}^{3} 3 θ - \text{sin} 3 θ \text{cos}^{2} 3 θ = 0$

for $0 ^{\circ} \leq θ < 180 ^{\circ}$.

Solve the equation

$5 \text{sin} θ + 2 \text{cos} θ = 3$

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

State the greatest value of $5 \text{sin} θ + 2 \text{cos} θ$ and the second positive value of $θ$ for which it occurs. Give your value of $θ$ correct to 3 significant figures.

Solve the equation

$3 \text{sec}^{4} θ + 16 = 16 + 16 \text{tan}^{2} θ$

for $- π \leq θ \leq π$, giving your answers in terms of $π$.

Given that

\text{cos} \left( A - B \right) = -\dfrac{\sqrt{3}}{2}

and

\text{tan} \left( \tfrac{1}{2}A - B \right) = \sqrt{3}

where $0 \leq 2 B < A \leq 360 ^{\circ}$, find the possible values of $A$ and $B$.

Solve the equation

$\text{cosec}^{2} x - 2 \frac{\text{cosec}x}{\text{sec}x} = 9$

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

Solve the equation

$8 \text{sin}^{4} 2 θ = 2 - 5 \text{cos} 4 θ$

for $- \frac{1}{2} π \leq θ \leq \frac{1}{2} π$, giving your answers in terms of $π$.

Given that $x = - 2$ is a root of the equation

$x^{3} + 12 x^{2} + 44 x + 48 = 0$

solve the equation by factorisation.

Hence solve the equation

\text{sec} \; \theta \left( \text{sec}^2 \theta + 44 \right) + 12 \left( \text{tan}^2 \theta + 5 \right) = 0

for $0 ^{\circ} \leq θ \leq 180 ^{\circ}$, giving your answers correct to 1 decimal place where appropriate.