Compound & Double Angle Formulae (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

Express tan $\left(210°\right)$ in terms of tan $\left(180°\right)$and tan $\left(30°\right)$

Hence show that tan$\left(210°\right)=\frac{\sqrt{3}}{3}$.

Starting with the identity

cos$\left(A+B\right)\equiv$cos $A$ cos $B$$-$sin $A$ sin $B$

and using the substitution $B=A$, show that cos $2A\equiv$cos² $A-$sin² $A$

Hence, or otherwise, show that cos $2A\equiv 1-2$ sin² $A$.

Using an appropriate trigonometric identity, show that 

$R\sin \left(\theta +\mathrm{\alpha }\right)\equiv R\cos \alpha \sin \theta +\mathrm{R}\sin \mathrm{\alpha }\cos \theta$

where $R$ and $\alpha$ are constants.

Hence show that $3\sin \theta +2\cos \theta =\sqrt{13}\sin \left(\theta +0.588\right)$.

Use appropriate double angle formulae to solve the following equations in the given intervals.

cos² $\theta -$sin² $\theta =\frac{1}{2}$    $-\mathrm{\pi }\le \theta \le \mathrm{\pi }$

$4\sin x\cos x=-\sqrt{3}0\le x\le \mathrm{\pi }$

Show that

$\frac{5\sin 2x}{\tan x}\equiv 10{\cos }^{2}xx\ne \frac{\mathrm{k\pi }}{2}$

(i) Show that $R\cos \left(x+\alpha \right)\equiv R\cos \alpha \cos x-R\sin \alpha \sin x$, where $R$ and $\alpha$ are constants.

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

Hence solve the equation  $\cos x-\sqrt{3}\sin x=1$ for  $0\le x\le 2\mathrm{\pi }$.

Using the identities

sin $\left(A+B\right)\equiv$sin $A$ cos $B+$sin $B$ cos $A$   and

cos $2A\equiv 1-2$ sin² $A$

show that sin $3A\equiv 3$ sin $A-4$ sin³ $A$

Hence, or otherwise, solve the equation

$3$ sin $\theta -4$sin³ $\theta =\frac{1}{2}-\mathrm{\pi }\le \theta \le \mathrm{\pi }$

Show that $5\sin \theta +12\cos \theta$ can be written in the form $R\sin \left(\theta +\mathrm{\alpha }°\right)$ where $R>0$ and $0°<\alpha <90°$.

Sketch the graph of $y=5$sin $x+12$cos $x$ for $0°\le x\le 360°$.

Label any points where the graph intercepts the coordinate axes.

Show that $2$ cosec $2A\equiv$cosec $A$ sec $A$.

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

$V=55\sqrt{3}\sin \frac{\pi t}{30}+55\cos \frac{\pi t}{30}.$

Show that

$55\sqrt{3}\sin \frac{\pi t}{30}+55\cos \frac{\pi t}{30}$

can be written as

$R\sin \left(\frac{\pi t}{30}+\alpha \right)$

where $R=110$and $\alpha =\frac{\mathrm{\pi }}{6}$.

(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?

If $A=B$, then

sin$\left(A-B\right)=$sin$\left(A-A\right)=$sin$\left(0\right)=0=$sin $A-$sin $A=$sin $A-$sin $B$

By using a suitable counter-example with $A\ne B$, prove that sin$\left(A-B\right)=$sin $A-$sin $B$ is not true in general.

Express cos$\left(285°\right)$ in terms of cosines and sines of 315° and 30°.

Hence show that cos$\left(285°\right)=\frac{\sqrt{6}-\sqrt{2}}{4}$.

Show that

sin $2A\equiv 2$ sin $A$ cos $A$

(You may use the identity sin $\left(A+B\right)\equiv$sin $A$ cos $B+$cos $A$ sin $B$.)

Show that  $2$ cos $\theta -5$ sin $\theta$  can be written in the form  $R$ cos$\left(\theta +\mathrm{\alpha }\right)$,  where  $R$and $\alpha$ are constants with   $R>0$ and  $0<\alpha <\frac{\mathrm{\pi }}{2}$  .

Give $R$ in the form  $\sqrt{k}$ where $k$ is an integer, and give $\alpha$ correct to three significant figures.

Solve the equation

sin $2\theta =$sin $\theta$             $-\mathrm{\pi }\le \mathrm{\theta }\le \mathrm{\pi }$

Solve the equation

cos $2x+$sin² $x=0$           $0\le x\le 2\mathrm{\pi }$

Show that

$\frac{\sin \left(A+B\right)+\sin \left(A-B\right)}{\cos \left(A+B\right)+\cos \left(A-B\right)}\equiv \tan A\left(A,B\ne \left(\mathrm{k}+\frac{1}{2}\right)\mathrm{\pi }\right)$

Show that $2\sin \theta +4\cos \theta$ can be written as $2\sqrt{5}\cos \left(\theta -\mathrm{\alpha }\right)$ , where $\alpha =0.464$ to three significant figures.

Hence solve the equation

$2$ sin $\theta +4$cos $\theta =3-\mathrm{\pi }\le \theta \le \mathrm{\pi }$

giving your answers correct to 3 significant figures.

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

Hence, or otherwise, solve the equation

$\frac{6\tan x-2{\tan }^{3}x}{1-3{\tan }^{2}x}=20\le x\le \mathrm{\pi }$

Sketch the graph of $y=2\left(\sin x-\cos x\right)$ for $0°\le x\le 360°$.

Be sure to label any points where the graph intercepts the coordinate axes, and state the coordinates of any maximum and minimum points.

Show that

$2-2$cot $2A$tan $A\equiv$sec² $AA\ne \mathrm{k\pi }$

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

$V=55\sqrt{2}\left(\sin \frac{\pi t}{60}+\cos \frac{\pi t}{60}\right).$

Express

$55\sqrt{2}\left(\sin \frac{\pi t}{60}+\cos \frac{\pi t}{60}\right).$

in the form

$R\sin \left(\frac{\pi t}{60}+\alpha \right)$

where $R$ and $\alpha$ are constants to be found.  $R>0$ and is 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 (cycle).

(i) Prove that $\sin \left(A-B\right)=\sin A+\sin B$  is not true in general.

(ii) Find values for $A$and $B$, with $A\ne 0$ and $B\ne 0$, for which    $\sin \left(A-B\right)=\sin A+\sin B$  is true.

Use the identities $\sin \left(A\pm B\right)\equiv \sin A\cos B\pm \cos A\sin B$ and $\cos \left(A\pm B\right)\equiv \cos A\cos B\pm \sin A\sin B$  to show that

      $$\begin{gathered} \sin \left(X+Y-Z\right)\equiv \\ \sin X\cos Y\cos Z+\cos X\sin Y\cos Z-\cos X\cos Y\sin Z+\sin X\sin Y\sin Z \end{gathered}$$

Hence show that $\sin \left(165°\right)=\frac{\sqrt{6}-\sqrt{2}}{4}$.

Show that

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

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

Given that  $a\sin \theta +\mathrm{b}\cos \theta$, where $a$ and $b$ are positive constants, is to be written in the form  $R\sin \left(\theta +\mathrm{\alpha }\right)$,  find expressions for:

(i) $\alpha$in terms of $a$ and $b$

(ii) $R$in terms of $a$ and $b$

Solve the equation          

$\cos 2\theta =\cos \theta 0\le \theta <2\mathrm{\pi }$

Solve the equation          

tan $2x=3$tan $x-\mathrm{\pi }\le x\le \mathrm{\pi }$

Show that

tan $2\theta$tan $\theta \equiv$sec $2\theta -1$

Show that $5\sin \theta -3\cos \theta$ can be written in the form  $R\sin \left(\theta -\mathrm{\alpha }\right)$ where    $R=\sqrt{34}$, and $\alpha =0.540$ radians correct to three significant figures.

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

$3\cos 2x+5\sin 2x=0.40\le x\le 2\mathrm{\pi }$

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

Hence, or otherwise, solve the equation

$2\cos 4x=7{\sin }^{2}x-20\le x\le \mathrm{\pi }$

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

Given that   $\alpha =A+B$, find the exact values of sin $\alpha$,cos $\alpha$ and tan $\alpha$.

(i) Explain briefly why $\theta =0$ is not a solution to the equation  $3\theta \cot 2\theta =0$.

(ii) By using an appropriate approximation, determine the value of $\underset{\theta \to 0}{\lim }3\theta \cot 2\theta$

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

$V=115\sin \omega t+115\sqrt{3}\cos \omega t.$

Express

$115\sin \omega t+115\sqrt{3}\cos \omega t$

in the form

$R\sin \left(\omega t+\alpha \right)$

where $R$ and $\alpha$are constants to be found. $R>0$and is $\alpha$ acute.

In the UK, domestic electricity runs at a frequency, $f$, of 50 Hertz (Hz).

The constant $\omega$ , is given by $\omega =2\pi f.$

(i) Find the initial voltage when a domestic appliance (such as a kettle or TV) 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.