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

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

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

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.

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

Hence show that

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

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

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

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

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.

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.

(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 $-\pi \le \theta \le \pi$, 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

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