Compound & Double Angle Formulae (Cambridge International AS Mathematics): Exam Questions

(i) Write down the exact value of $\text{cos} 60 ^{\circ}$.

(ii) Write down the exact value of $\text{cos} 45 ^{\circ}$.

(iii) Use the compound angle formula for $\text{cos} ( A + B )$ to find the exact value of $\text{cos} 105 ^{\circ}$.

(iv) Hence show that $\text{cos} 60 ^{\circ} + \text{cos} 45 ^{\circ} \neq \text{cos} 105 ^{\circ}$.

Express $\text{sin} 15 ^{\circ}$ in terms of sines and cosines of $45 ^{\circ}$ and $30 ^{\circ}$.

Hence show that

$\text{sin} 15 ^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}$

Starting with the identity

$\text{sin} ( A + B ) \equiv \text{sin} A \text{cos} B + \text{sin} B \text{cos} A$

and using the substitution $B = A$, show that $\text{sin} 2 A \equiv 2 \text{sin} A \text{cos} A$.

Hence show that $\text{sin} 120 ^{\circ} = \frac{\sqrt{3}}{2}$.

Use an appropriate identity to find $\text{sin} ( θ + α )$ in terms of sines and cosines of $θ$ and $α$.

Hence show that $R \text{sin} ( θ + α ) \equiv R \text{cos} α \text{sin} θ + R \text{sin} α \text{cos} θ$.

Solve the equation

$\text{sin} 2 θ = \frac{1}{2}$

for $- π \leq θ \leq π$.

Solve the equation

$\text{cos} 2 θ = \frac{\sqrt{3}}{2}$

for $0 \leq θ \leq 2 π$.

Show that

5 \; \text{cos}\left(\theta - \dfrac{\pi}{6}\right) \equiv \dfrac{5\sqrt{3}}{2} \; \text{cos} \; \theta + \dfrac{5}{2} \; \text{sin} \; \theta

Show that

$\text{cos}^{2} x + \text{cos} 2 x \equiv 3 \text{cos}^{2} x - 1$

Sketch the graph of $y = \text{tan} 2 θ$ for $0 \leq θ \leq 2 π$, showing the coordinates of the points of intersection with the axes.

Use the difference of two squares to show that

$\text{cos}^{4} x - \text{sin}^{4} x \equiv \text{cos} 2 x$

Hence solve the equation

$\text{cos}^{4} x - \text{sin}^{4} x = \frac{\sqrt{2}}{2}$

for $- \frac{π}{2} \leq x \leq \frac{π}{2}$.

The alternating voltage, $V$, in an electrical circuit, $t$ seconds after it is switched on, is modelled by the function

$V = 20 \text{cos} π t + 20 \sqrt{3} \text{sin} π t$

Use the identity $R \text{cos} ( π t - α ) \equiv R \text{cos} α \text{cos} π t + R \text{sin} α \text{sin} π t$ to show that

$20 \text{cos} π t + 20 \sqrt{3} \text{sin} π t$

can be written as

40 \; \text{cos}\left(\pi t - \dfrac{\pi}{3}\right)

(i) Write down the maximum voltage in the electrical circuit.

(ii) Find the voltage at time $t = 0$.

(iii) Find the voltage after two seconds.

After how many seconds does the voltage first equal $- 20$ volts?

Express $\text{tan} 210 ^{\circ}$ in terms of $\text{tan} 180 ^{\circ}$ and $\text{tan} 30 ^{\circ}$.

Hence show that $\text{tan} 210 ^{\circ} = \frac{\sqrt{3}}{3}$.

Starting with the identity

$\text{cos} ( A + B ) \equiv \text{cos} A \text{cos} B - \text{sin} A \text{sin} B$

and using the substitution $B = A$, show that $\text{cos} 2 A \equiv \text{cos}^{2} A - \text{sin}^{2} A$.

Hence, or otherwise, show that $\text{cos} 2 A \equiv 1 - 2 \text{sin}^{2} A$.

Using an appropriate trigonometric identity, show that

$R \text{sin} ( θ + α ) \equiv R \text{cos} α \text{sin} θ + R \text{sin} α \text{cos} θ$

where $R$ and $α$ are constants.

Hence show that $3 \text{sin} θ + 2 \text{cos} θ = \sqrt{13} \text{sin} ( θ + 0 . 588 )$.

By using an appropriate double angle formula, solve the equation

$\text{cos}^{2} θ - \text{sin}^{2} θ = \frac{1}{2}$

for $- π \leq θ \leq π$.

By using an appropriate double angle formula, solve the equation

$4 \text{sin} x \text{cos} x = - \sqrt{3}$

for $0 \leq x \leq π$.

Show that

$\frac{5\text{sin}2x}{\text{tan}x} \equiv 10 \text{cos}^{2} x$

for $x \neq \frac{kπ}{2}$.

(i) Show that $R \text{cos} ( x + α ) \equiv R \text{cos} α \text{cos} x - R \text{sin} α \text{sin} x$, where $R$ and $α$ are constants.

(ii) Use your result from part (i) to show that \text{cos} \; x - \sqrt{3} \; \text{sin} \; x \equiv 2 \; \text{cos}\left(x + \dfrac{\pi}{3}\right).

Hence solve the equation $\text{cos} x - \sqrt{3} \text{sin} x = 1$ for $0 \leq x \leq 2 π$.

Using the identities

$\text{sin} ( A + B ) \equiv \text{sin} A \text{cos} B + \text{sin} B \text{cos} A$

and

$\text{cos} 2 A \equiv 1 - 2 \text{sin}^{2} A$

show that $\text{sin} 3 A \equiv 3 \text{sin} A - 4 \text{sin}^{3} A$.

Hence, or otherwise, solve the equation

$3 \text{sin} θ - 4 \text{sin}^{3} θ = \frac{1}{2}$

for $- π \leq θ \leq π$.

Show that $5 \text{sin} θ + 12 \text{cos} θ$ can be written in the form $R \text{sin} ( θ + α ^{\circ} )$, where $R > 0$ and $0 ^{\circ} < α < 90 ^{\circ}$.

Sketch the graph of $y = 5 \text{sin} x + 12 \text{cos} x$ for $0 ^{\circ} \leq x \leq 360 ^{\circ}$, showing the coordinates of the points of intersection with the axes.

Show that

$2 \text{cosec} 2 A \equiv \text{cosec} A \text{sec} A$

The alternating voltage, $V$, in an electrical circuit, $t$ seconds after it is switched on, is modelled by the function

$V = 55 \sqrt{3} \text{sin} \frac{πt}{30} + 55 \text{cos} \frac{πt}{30}$

Show that

$55 \sqrt{3} \text{sin} \frac{πt}{30} + 55 \text{cos} \frac{πt}{30}$

can be written as

R \; \text{sin}\left(\dfrac{\pi t}{30} + \alpha\right)

where $R = 110$ and $α = \frac{π}{6}$.

(i) Find the voltage at time $t = 0$.

(ii) Find the voltage after one minute.

After how many seconds does the voltage first equal $- 55$ volts?

(i) Show that $R \text{sin} ( θ + α ) \equiv R \text{cos} α \text{sin} θ + R \text{sin} α \text{cos} θ$, where $R$ and $α$ are constants with $R > 0$ and $0 < α < \frac{π}{2}$.

(ii) Use your result from part (i) to show that \sqrt{3} \; \text{sin} \; \theta + \text{cos} \; \theta \equiv 2 \; \text{sin}\left(\theta + \dfrac{\pi}{6}\right).

Write down the maximum value of $\sqrt{3} \text{sin} θ + \text{cos} θ$.

If $A = B$, then

$\text{sin} ( A - B ) = \text{sin} ( A - A ) = \text{sin} 0 = 0 = \text{sin} A - \text{sin} A = \text{sin} A - \text{sin} B$

By using a suitable counter-example with $A \neq B$, show that $\text{sin} ( A - B ) = \text{sin} A - \text{sin} B$ does not hold for all values of $A$ and $B$.

Express $\text{cos} 285 ^{\circ}$ in terms of cosines and sines of $315 ^{\circ}$ and $30 ^{\circ}$.

Hence show that $\text{cos} 285 ^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}$.

Show that

$\text{sin} 2 A \equiv 2 \text{sin} A \text{cos} A$

You may use the identity $\text{sin} ( A + B ) \equiv \text{sin} A \text{cos} B + \text{cos} A \text{sin} B$.

Show that $2 \text{cos} θ - 5 \text{sin} θ$ can be written in the form $R \text{cos} ( θ + α )$, where $R$ and $α$ are constants with $R > 0$ and $0 < α < \frac{π}{2}$.

Give $R$ in the form $\sqrt{k}$, where $k$ is an integer, and give $α$ correct to three decimal places.

Show that $2 \text{sin} θ + 4 \text{cos} θ$ can be written as $2 \sqrt{5} \text{cos} ( θ - α )$, where $α = 0 . 464$ correct to three decimal places.

Hence solve the equation

$2 \text{sin} θ + 4 \text{cos} θ = 3$

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

The alternating voltage, $V$, in an electrical circuit, $t$ seconds after it is switched on, is modelled by the function

V = 55\sqrt{2}\left(\text{sin} \; \dfrac{\pi t}{60} + \text{cos} \; \dfrac{\pi t}{60}\right)

Express

55\sqrt{2}\left(\text{sin} \; \dfrac{\pi t}{60} + \text{cos} \; \dfrac{\pi t}{60}\right)

in the form

R \; \text{sin}\left(\dfrac{\pi t}{60} + \alpha\right)

where $R$ and $α$ are constants to be found, with $R > 0$ and $α$ acute.

Find the voltage when the circuit is switched on.

(i) Write down the maximum voltage and the time at which this first occurs.

(ii) Find the time it takes the voltage to complete one period.

Solve the equation

$\text{sin} 2 θ = \text{sin} θ$

for $- π \leq θ \leq π$.

Solve the equation

$\text{cos} 2 x + \text{sin}^{2} x = 0$

for $0 \leq x \leq 2 π$.

Show that

$\frac{\text{sin}(A+B)+\text{sin}(A-B)}{\text{cos}(A+B)+\text{cos}(A-B)} \equiv \text{tan} A$

for A, \; B \neq \left(k + \dfrac{1}{2}\right)\pi.

By letting $B = 2 A$, use the identity for $\text{tan} ( A + B )$ to derive an expression for $\text{tan} 3 A$ in terms of $\text{tan} A$.

Hence, or otherwise, solve the equation

$\frac{6\text{tan}x-2\text{tan}^{3}x}{1-3\text{tan}^{2}x} = 2$

for $0 \leq x \leq π$.

Sketch the graph of y = 2\left(\text{sin} \; x - \text{cos} \; x\right) for $0 ^{\circ} \leq x \leq 360 ^{\circ}$.

Label any points where the graph intercepts the coordinate axes, and state the coordinates of any maximum and minimum points.

Show that

$2 - 2 \text{cot} 2 A \text{tan} A \equiv \text{sec}^{2} A$

for $A \neq \frac{kπ}{2}$.

(i) Show that $\text{sin} ( A - B ) = \text{sin} A + \text{sin} B$ does not hold for all values of $A$ and $B$.

(ii) Find values for $A$ and $B$, with $A \neq 0$ and $B \neq 0$, for which the statement does hold.

Use the identities

$\text{sin} ( A \pm B ) \equiv \text{sin} A \text{cos} B \pm \text{cos} A \text{sin} B$

$\text{cos} ( A + B ) \equiv \text{cos} A \text{cos} B - \text{sin} A \text{sin} B$

$\text{cos} ( A - B ) \equiv \text{cos} A \text{cos} B + \text{sin} A \text{sin} B$

to show that

$\text{sin} ( X + Y - Z ) \equiv \text{sin} X \text{cos} Y \text{cos} Z + \text{cos} X \text{sin} Y \text{cos} Z - \text{cos} X \text{cos} Y \text{sin} Z + \text{sin} X \text{sin} Y \text{sin} Z$

Hence show that $\text{sin} 165 ^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}$.

Show that

$\text{tan} 2 A \equiv \frac{2\text{tan}A}{1-\text{tan}^{2}A}$

State clearly any trigonometric identities you use to show this result.

Given that $a \text{sin} θ + b \text{cos} θ$, where $a$ and $b$ are positive constants, is to be written in the form $R \text{sin} ( θ + α )$, find expressions for:

(i) $α$ in terms of $a$ and $b$

(ii) $R$ in terms of $a$ and $b$

Solve the equation

$\text{cos} 2 θ = \text{cos} θ$

for $0 \leq θ < 2 π$.

Solve the equation

$\text{tan} 2 x = 3 \text{tan} x$

for $- π \leq x \leq π$.

Show that

$\text{tan} 2 θ \text{tan} θ \equiv \text{sec} 2 θ - 1$

The diagram below shows two right-angled triangles. Angles $A$ and $B$ have been labelled.

Given that $α = A + B$, find the exact values of $\text{sin} α$, $\text{cos} α$ and $\text{tan} α$.

The alternating voltage, $V$, in a domestic electrical circuit, $t$ seconds after it is switched on, is modelled by the function

$V = 115 \text{sin} ω t + 115 \sqrt{3} \text{cos} ω t$

Express

$115 \text{sin} ω t + 115 \sqrt{3} \text{cos} ω t$

in the form

$R \text{sin} ( ω t + α )$

where $R$ and $α$ are constants to be found, with $R > 0$ and $α$ acute.

In the UK, domestic electricity runs at a frequency, $f$, of 50 Hertz (Hz). The constant $ω$ is given by $ω = 2 π f$.

(i) Find the initial voltage when a domestic appliance, such as a kettle or a television, is switched on.

(ii) Find the time at which the voltage first turns negative.

(i) Find the period of one cycle of voltage in the UK.

(ii) In the US, the period of one cycle is $\frac{1}{60}$ seconds. Write down the frequency of US domestic electricity.

Show that $5 \text{sin} θ - 3 \text{cos} θ$ can be written in the form $R \text{sin} ( θ - α )$, where $R = \sqrt{34}$ and $α = 0 . 540$ radians correct to three decimal places.

Use your result from part (a), and the properties of the sine and cosine functions, to solve the equation

$3 \text{cos} 2 x + 5 \text{sin} 2 x = 0 . 4$

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

Use an identity for $\text{cos} 2 A$ to derive an identity for $\text{cos} 4 A$, in terms of $\text{cos} A$.

Hence, or otherwise, solve the equation

$2 \text{cos} 4 x = 7 \text{sin}^{2} x - 2$

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