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

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

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 $A$.

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 $AA\ne \mathrm{k\pi }$

(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 cos $A$.

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 $\underset{\theta \to 0}{\lim }3\theta \cot 2\theta$