Compound & Double Angle Formulae (Edexcel IAL Maths): Exam Questions

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

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

(iii) Use your calculator to find the exact value of cos $105^{\circ}$.

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

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

Hence show that

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

Starting with the identity

sin$( A + B ) \equiv$sin $A$cos $B+$sin $B$cos $A$

And using the substitution $B=A$, show that sin $2A\equiv2$sin $A$ cos $A$.

Hence show the exact value of sin $120^{\circ}=\frac{\sqrt{3}}{2}$.

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

Hence show that $R$ sin$( \\theta + \ \\alpha ) \equiv R$ cos $α$ sin $θ+R$ sin $α$ cos $θ$.

Solve the following equations in the given intervals.

sin $2θ=\frac{1}{2},-π\leqθ\leqπ$

cos $2θ=\frac{\sqrt{3}}{2},0\leqθ\leq2π$

Show that

5 cos ( theta - pi / 6 ) === (5 sqrt( 3)/(2) cos theta + 5 / 2 sin theta

Show that

$\mathrm{cos}^{2}x+\mathrm{cos}2x\equiv3\mathrm{cos}^{2}x-1$

(i) Show that , $R\mathrm{sin}\quad ( \\theta + \ \\alpha ) \equiv R$ $\mathrm{cos}α$ $\mathrm{sin}θ+R$ $\mathrm{sin}α$ $\mathrm{cos}θ$ 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 ) sin theta + cos theta === 2 sin ( theta + pi / 6 ).

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

Sketch the graph of $y=\mathrm{tan}2θ$ for $0\leqθ\leq2π$.

Label the points at which the graph intersects the coordinate axes.

Use the difference of two squares to show that

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

Hence solve the equation

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

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

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

Express tan $( 210^{\circ} )$ in terms of tan $( 180^{\circ} )$and tan $( 30^{\circ} )$

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

Starting with the identity

cos$( A + B ) \equiv$cos $A$ cos $B\quad -$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 $2A\equiv1-2$ sin² $A$.

Using an appropriate trigonometric identity, show that

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

where $R$ and $α$ are constants.

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

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

cos² $θ-$sin² $θ=\frac{1}{2}$ $-π\leqθ\leqπ$

$4\mathrm{sin}x\mathrm{cos}x=-\sqrt{3}0\leq x\leqπ$

Show that

$\frac{5\mathrm{sin}2x}{\mathrm{tan}x}\equiv10\mathrm{cos}^{2}xx\neq\frac{\mathrm{kπ}}{2}$

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

(ii) Use your result from part (i) to show that cos x - sqrt( 3 sin x === 2 cos ( x + pi / 3 ).

Hence solve the equation cos $x-\sqrt{3}$sin $x=1$ for $0\leq x\leq2π$.

Using the identities

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

cos $2A\equiv1-2$ sin² $A$

show that sin $3A\equiv3$ sin $A-4$ sin³ $A$

Hence, or otherwise, solve the equation

$3$ sin $θ-4$sin³ $θ=\frac{1}{2}-π\leqθ\leqπ$

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

Sketch the graph of $y=5$sin $x+12$cos $x$ for $0^{\circ}\leq x\leq360^{\circ}$. 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$( A - B ) =$sin$( A - A ) =$sin$( 0 ) = 0 =$sin $A-$sin $A=$sin $A-$sin $B$

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

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

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

Show that

sin $2A\equiv2$ sin $A$ cos $A$

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

Show that $2$ cos $θ-5$ sin $θ$ can be written in the form $R$ cos$( \\theta + \ \\alpha )$, 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 significant figures.

Solve the equation

sin $2θ=$sin $θ$ $-π\leqθ\leqπ$

Solve the equation

cos $2x+$sin² $x=0$ $0\leq x\leq2π$

Show that

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

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

Hence solve the equation

$2$ sin $θ+4$cos $θ=3-π\leqθ\leqπ$

giving your answers correct to 3 significant figures.

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

Hence, or otherwise, solve the equation

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

Sketch the graph of $y = 2 ( \sin x - \cos x )$ for $0^{\circ}\leq x\leq360^{\circ}$.

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 $AA\neq\mathrm{kπ}$

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

(ii) Find values for $A$and $B$, with $A\neq0$ and $B\neq0$, for which $\sin ( A - B ) = \sin A + \sin B$ is true.

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

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

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

Show that

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

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

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

(i) $α$in terms of $a$ and $b$

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

Solve the equation

$\mathrm{cos}2θ=\mathrm{cos}θ0\leqθ<2π$

Solve the equation

tan $2x=3$tan $x-π\leq x\leqπ$

Show that

tan $2θ$tan $θ\equiv$sec $2θ-1$

Show that $5\mathrm{sin}θ-3\mathrm{cos}θ$ can be written in the form $R \sin ( \\theta - \ \\alpha )$ where $R=\sqrt{34}$, and $α=0.540$ 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

$3\mathrm{cos}2x+5\mathrm{sin}2x=0.40\leq x\leq2π$

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

Hence, or otherwise, solve the equation

$2\mathrm{cos}4x=7\mathrm{sin}^{2}x-20\leq x\leqπ$

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 sin $α$,cos $α$ and tan $α$.

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

(ii) By using an appropriate approximation, determine the value of $\underset{θ\rightarrow0}{\mathrm{lim}}3θ\mathrm{cot}2θ$