Compound & Double Angle Formulae (Edexcel A Level Maths: Pure): Exam Questions

Solve, for $-\pi \le \theta \le \pi$, the equation

$\text{sin}2\theta =\frac{1}{2}$

Solve, for $0\le \theta \le 2\pi$, the equation

$\text{cos}2\theta =\frac{\sqrt{3}}{2}$

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use the difference of two squares to show that

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

Hence solve, for $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$, the equation

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

Solve, for $0\le x<\frac{\pi }{2}$, the equation

$4\sin x=\sec x$

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

$\cos 3A\equiv 4{\cos }^{3}A-3\cos A$

Hence solve, for $-90°\le x\le 180°$, the equation

$1-\cos 3x={\sin }^{2}x$

In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Given that $1+\cos 2\theta +\sin 2\theta \ne 0$ prove that

$\frac{1-\cos 2\theta +\sin 2\theta }{1+\cos 2\theta +\sin 2\theta }\equiv \tan \theta$

Hence solve, for $0<x<180°$

$\frac{1-\cos 4x+\sin 4x}{1+\cos 4x+\sin 4x}=3\sin 2x$

giving your answers to one decimal place where appropriate.

Prove

$\frac{\cos 3\theta }{\sin \theta }+\frac{\sin 3\theta }{\cos \theta }\equiv 2\cot 2\theta \theta \ne \left(90n\right)°,n\in \mathrm{\mathbb{Z}}$

Hence solve, for $90°<\theta <180°$, the equation

$\frac{\cos 3\theta }{\sin \theta }+\frac{\sin 3\theta }{\cos \theta }=4$

giving any solutions to one decimal place.

Express $2\cos \theta +8\sin \theta$ in the form $R\cos \left(\theta -\alpha \right)$, where $R$ and $\alpha$ are constants, $R>0$ and $0<\alpha <\frac{\pi }{2}$.

Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 decimal places.

The first three terms of an arithmetic sequence are

$\cos x\cos x+\sin x\cos x+2\sin xx\ne n\pi$

Given that $S_9$ represents the sum of the first 9 terms of this sequence as $x$ varies,

(i) find the exact maximum value of $S_9$

(ii) deduce the smallest positive value of $x$ at which this maximum value of $S_9$ occurs.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

By writing $210°$ as $(180°+30°)$, express $\text{tan}210°$ in terms of $\text{tan}180°$ and $\text{tan}30°$.

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

Show that

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

where $R$ and $\alpha$ are constants.

Hence show that

$3\text{sin}\theta +2\text{cos}\theta \equiv \sqrt{13}\text{sin}\left(\theta +0.588\right)$

where $0.588$ is measured in radians to 3 decimal places.

Express $3\cos \theta +4\sin \theta$ in the form $R\cos \left(\theta -\alpha \right)$, where $R>0$ and $0<\alpha <\frac{\pi }{2}$.

Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 decimal places.

Hence solve the equation $3\cos \theta +4\sin \theta =2.5$ for $0\le \theta <2\pi$.

Write your answers to 3 significant figures.

Solve, for $-\pi \le \theta \le \pi$, the equation

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

Solve, for $0\le x\le \pi$, the equation

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

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

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

(i) Show that

$R\text{cos}\left(x+\alpha \right)\equiv R\text{cos}\alpha \text{cos}x-R\text{sin}\alpha \text{sin}x$

where $R$ and $\alpha$ are constants.

(ii) Hence show that

$\text{cos}x-\sqrt{3}\text{sin}x\equiv 2\text{cos}\left(x+\frac{\pi }{3}\right)$

Hence solve, for $0\le x\le 2\pi$, the equation

$\text{cos}x-\sqrt{3}\text{sin}x=1$

Express $5\text{sin}\theta +12\text{cos}\theta$ in the form $R\text{sin}\left(\theta +\alpha °\right)$, where $R>0$ and $0°<\alpha <90°$.

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

Show on your sketch the coordinates of the points where the graph crosses the coordinate axes.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

$2\text{cosec}2A\equiv \text{cosec}A\text{sec}AA\ne \frac{k\pi }{2}$

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

By writing $285°$ as $(315°-30°)$, express $\text{cos}285°$ in terms of the sine and cosine of $315°$ and $30°$.

Hence show that

$\text{cos}\left(285°\right)=\frac{\sqrt{6}-\sqrt{2}}{4}$

Express $2\text{cos}\theta -5\text{sin}\theta$ in the form $R\text{cos}\left(\theta +\alpha \right)$, where $R>0$ and $0<\alpha <\frac{\pi }{2}$.

Give the exact value of $R$, and give the value of $\alpha$ in radians correct to 3 significant figures.

(i) Disprove the following statement by means of a counter example:

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

(ii) Find a value for $A$ and a value for $B$, where $A\ne 0$ and $B\ne 0$, such that

$\text{sin}\left(A-B\right)=\text{sin}A+\text{sin}B$

By writing $2A$ as $(A+A)$ show that

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

You must clearly state any trigonometric identities you use in your proof.

Given that $a$ and $b$ are positive constants, and that

$a\text{sin}\theta +b\text{cos}\theta \equiv R\text{sin}\left(\theta +\alpha \right)$

where $R>0$ and $0<\alpha <\frac{\pi }{2}$,

(i) find an expression for $\alpha$ in terms of $a$ and $b$,

(ii) find an expression for $R$ in terms of $a$ and $b$.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

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

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

$2-2\text{cot}2A\text{tan}A\equiv {\text{sec}}^{2}AA\ne k\pi$

By using the identity for $\text{sin}\left(A+B\right)$ and the substitution $\text{cos}2A\equiv 1-2{\text{sin}}^{2}A$, show that

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

Hence solve, for $-\pi \le \theta \le \pi$, the equation

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

Solve, for $-180°\le \theta \le 180°$, the equation

$5\sin 2\theta =9\tan \theta$

giving your answers, where necessary, to one decimal place.

[Solutions based entirely on graphical or numerical methods are not acceptable.]

Deduce the smallest positive solution to the equation

$5\sin \left(2x-50°\right)=9\tan \left(x-25°\right)$

Given that

$2\sin (x-60°)=\cos (x-30°)$

show that

$\tan x=3\sqrt{3}$

Hence or otherwise solve, for $0\le \theta <180°$

$2\sin 2\theta =\cos (2\theta +30°)$

giving your answers to one decimal place.

Express $2\cos \theta -\sin \theta$ in the form $R\cos \left(\theta +\alpha \right)$, where $R>0$ and $0<\alpha <\frac{\pi }{2}$

Give the exact value of $R$ and the value of $\alpha$ in radians to 3 decimal places.

Figure 6 shows the cross-section of a water wheel.

The wheel is free to rotate about a fixed axis through the point $C$.

The point $P$ is at the end of one of the paddles of the wheel, as shown in Figure 6.

The water level is assumed to be horizontal and of constant height.

The vertical height, $H$ metres, of $P$ above the water level is modelled by the equation

$H=3+4\cos \left(0.5t\right)-2\sin \left(0.5t\right)$

where $t$ is the time in seconds after the wheel starts rotating.

Using the model, find

(i) the maximum height of $P$ above the water level,

(ii) the value of $t$ when this maximum height first occurs, giving your answer to one decimal place.

In a single revolution of the wheel, $P$ is below the water level for a total of $T$ seconds.

According to the model, find the value of $T$ giving your answer to 3 significant figures.

(Solutions based entirely on calculator technology are not acceptable.)

Express $\sin x+2\cos x$ in the form $R\sin \left(x+\alpha \right)$ where $R$ and $\alpha$ are constants, $R>0$ and $0<\alpha <\frac{\pi }{2}$.

Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 decimal places.

The temperature, $\theta$ °C, inside a room on a given day is modelled by the equation

$\theta =5+\sin \left(\frac{\pi t}{12}-3\right)+2\cos \left(\frac{\pi t}{12}-3\right)0\le t<24$

where $t$ is the number of hours after midnight.

Using the equation of the model and your answer to part (a), deduce the maximum temperature of the room during this day.

Find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

Solve, for $0\le \theta <360°$, the equation

$5\sin \theta -5\cos \theta =2$

giving your answers to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

Solve, for $-\pi \le \theta \le \pi$, the equation

$\text{sin}2\theta =\text{sin}\theta$

Solve, for $0\le x\le 2\pi$, the equation

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

Express $2\text{sin}\theta +4\text{cos}\theta$ in the form $R\text{cos}\left(\theta -\alpha \right)$, where $R>0$ and $0<\alpha <\frac{\pi }{2}$.

Give the exact value of $R$, and give the value of $\alpha$ in radians to 3 significant figures.

Hence solve, for $-\pi \le \theta \le \pi$, the equation

$2\text{sin}\theta +4\text{cos}\theta =3$

giving your answers to 3 significant figures.

By writing $3A$ as $(2A+A)$ and using the identity for $\text{tan}\left(A+B\right)$, show that

$\text{tan}3A\equiv \frac{3\text{tan}A-{\text{tan}}^{3}A}{1-3{\text{tan}}^{2}A}$

Hence solve, for $0\le x\le \pi$, the equation

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

(i) Express $2\text{sin}x-2\text{cos}x$ in the form $R\text{sin}\left(x-\alpha °\right)$, where $R>0$ and $0°<\alpha <90°$.

Give the exact value of $R$ and the value of $\alpha$.

(ii) Hence sketch the curve with equation

$y=2\left(\text{sin}x-\text{cos}x\right)0°\le x\le 360°$

Show on your sketch the coordinates of the points where the curve crosses the coordinate axes, and state the exact coordinates of the maximum and minimum turning points.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

By writing $\left(X+Y-Z\right)$ as $\left(\left(X+Y\right)-Z\right)$ and using the identities for $\text{sin}\left(A\pm B\right)$ and $\text{cos}\left(A\pm B\right)$, show that

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

Hence show that

$\text{sin}165°=\frac{\sqrt{6}-\sqrt{2}}{4}$

Solve, for $0\le \theta <2\pi$, the equation

$\text{cos}2\theta =\text{cos}\theta$

Solve, for $-\pi \le x\le \pi$, the equation

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

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

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

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

Hence or otherwise, solve for $0\le x\le 2\pi$, the equation

$3\text{cos}2x+5\text{sin}2x=0.4$

By using the double angle identity for $\text{cos}2A$, show that $\text{cos}4A$ can be expressed in the form

$a{\text{cos}}^{4}A+b{\text{cos}}^{2}A+c$

where $a$, $b$ and $c$ are constants to be found.

Hence solve, for $0\le x\le \pi$, the equation

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

Two right-angled triangles are shown in the diagram below. Angles $A$ and $B$ have been labelled.

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

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

$3\theta \text{cot}2\theta =0$

(ii) Given that $\theta$ is small and measured in radians, use the small angle approximations to find the value of

$3\theta \text{cot}2\theta$