Trigonometric Formulae & Identities (Edexcel IGCSE Further Pure Maths): Exam Questions

Figure 2 shows a right pyramid $ABCDO$with a horizontal square base of side 8 cm. The vertical height of the pyramid is $h$ cm and $OA=OB=OC=OD=12$ cm.

Find the exact value of $h$.

Find, to 1 decimal place, the size of the angle between $OA$ and the plane $ABCD$.

Find, to 1 decimal place, the size of the angle between the plane $AOB$ and the plane $ABCD$.

The midpoint of $OA$is $P$ and $Q$ is the point on $BC$ such that $BQ:QC=3:1$

Show that $PQ=4\sqrt{5}$ cm.

Find, to 1 decimal place, the size of angle $PQA$.

Show that $\cos (A–B)–\cos (A+B)=2\sin A\sin B$

Hence express $2\sin 5x\sin 3x$ in the form $\cos mx–\cos nx$where $m$ and $n$ are integers, giving the value of $m$ and the value of $n$,

(i) Find $\int 4\sin 5\theta \sin 3\theta \mathrm{d}\theta$

(ii) Hence evaluate ${\int }_0^{\frac{\mathrm{\pi }}{6}}4\sin 5\theta \sin 3\theta \mathrm{d}\theta$, giving your answer in the form $\frac{a\sqrt{b}}{c}$ where $a$, $b$ and $c$ are integers.

In triangle $ABC$, $AB$= $2x$cm, $BC=3x$ cm and $AC=4x$ cm

The area of triangle $ABC$ is $50$ cm²

Find, to 2 decimal places, the value of $x$

Figure 5 shows a right triangular prism $ABCDEF$where $ABCD$ is a rectangle.

$AF=DE\hspace{1em}\hspace{1em}BF=CE\hspace{1em}\hspace{1em}AD=FE=BC\hspace{1em}\hspace{1em}AB=DC=24$ cm

$\angle ABF=\angle DCE=45°\hspace{1em}\hspace{1em}\angle BAF=\angle CDE=60°$

Using a formula from page 2,

show that $\sin AFB=\frac{\sqrt{2}+\sqrt{6}}{4}$

Without using a calculator,

show that $BF=12\left(3\sqrt{2}-\sqrt{6}\right)$cm

The angle between the plane $AEB$and the plane $ABCD$is $65°$

Find, in cm to 2 significant figures, the length of $EF$

Find, in degrees to one decimal place, the size of the angle between the line $CF$and the plane $ABCD$

Using formulae (opens in a new tab), show that

(i) $\cos 2A=2{\cos }^{2}A-1$

(3)

(ii) $\sin 2A=2\sin A\cos A$

(1)

Show that ${\cos }^{3}A=\frac{\cos 3A+3\cos A}{4}$

Hence, or otherwise, solve, giving exact values in terms of $\pi$

$8{\cos }^3\left(\frac{\theta }{2}\right)-6\cos \left(\frac{\theta }{2}\right)-1=0$ for $0⩽\theta ⩽2\mathrm{\pi }$

use algebraic integration to find the exact value of

${\int }_0^{\frac{\mathrm{\pi }}{6}}\left(4{\cos }^3\theta -\sin 2\theta \right)\mathrm{d}\theta$

Triangle $ABC$is such that

$AC=10\mathrm{cm}BC=7\mathrm{cm}\hspace{1em}\hspace{1em}\mathrm{angle}CAB=25°$

Given that angle $ABC$is obtuse,

find, in cm to one decimal place, the length of $AB$

Using a formula (opens in a new tab), show that

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

Hence, solve the equation

$\tan A°-\tan 2A°=0$ for $0⩽A⩽180$

Using a formula given on page 2, solve, giving your solutions as exact values

$\cos \left(x-\frac{\pi }{6}\right)=\sin x$ for $-\pi ⩽x⩽2\pi$

Figure 2 shows triangle $AOB$

$\overrightarrow{OA}=4\mathbf{a}+5\mathbf{b}\overrightarrow{OB}=8\mathbf{a}-\mathbf{b}\overrightarrow{OD}=15\mathbf{a}+10\mathbf{b}$ where $\left|\mathbf{a}\right|=\left|\mathbf{b}\right|=1$

(i) Find in terms $\overrightarrow{AB}$ of $\mathbf{a}$ and $\mathbf{b}$

[2]

(ii) Find, in its simplest form, the exact value of $\left|\overrightarrow{AB}\right|$

[1]

Figure 2 shows triangle $AOB$

$\overrightarrow{OA}=4\mathbf{a}+5\mathbf{b}\overrightarrow{OB}=8\mathbf{a}-\mathbf{b}\overrightarrow{OD}=15\mathbf{a}+10\mathbf{b}$ where $\left|\mathbf{a}\right|=\left|\mathbf{b}\right|=1$

Find the area of triangle $AOB$

Figure 2 shows triangle $AOB$

$\overrightarrow{OA}=4\mathbf{a}+5\mathbf{b}\overrightarrow{OB}=8\mathbf{a}-\mathbf{b}\overrightarrow{OD}=15\mathbf{a}+10\mathbf{b}$ where $\left|\mathbf{a}\right|=\left|\mathbf{b}\right|=1$

The point $C$ lies on $AB$ and $OD$ such that $O$, $C$ and $D$ are collinear.

Use a vector method to find vector $\overrightarrow{OC}$ as a simplified expression in terms of $\mathbf{a}$ and $\mathbf{b}$

Using a formula (opens in a new tab), show that

$\cos 2\theta =2{\cos }^2\theta -1$

Using a formula (opens in a new tab), show that

$\cos 2\theta =2{\cos }^2\theta -1$

Hence show that

${\int }_{\frac{\pi }{3}}^{\frac{3\pi }{4}}\left(2{\cos }^2\theta -1\right) d\theta =-\frac{a+\sqrt{b}}{c}$

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

Figure 3 shows part of the curve $C_1$ with equation $y=2{\cos }^2\theta -1$and part of the curve $C_2$ with equation $y=-\cos \theta$

Point $B$ is the intersection of $C_1$ and $C_2$ as shown in Figure 3

Point $A$$\left(\frac{3\mathrm{\pi }}{4},0\right)$ is the intersection of $C_1$ with the $\theta$-axis as shown in Figure 3

Point $E$ $\left(\frac{\pi }{2},0\right)$ is the intersection of $C_2$ with the $\theta$-axis as shown in Figure 3

The finite region $R$, shown shaded in Figure 3, is bounded by the $\theta$-axis, $C_1$ and $C_2$

Use calculus to find, in its simplest form, the exact area of $R$

Using the formulae (opens in a new tab) , show that

(i) ${\cos }^{2}A=\frac{\cos 2A+1}{2}$

(ii) ${\sin }^{2}A=\frac{1-\cos 2A}{2}$

Show that

$(2\sin x-\cos x)(\sin x-3\cos x)=\frac{1}{2}(\cos 2x-7\sin 2x+5)$

$y=(2\sin x-\cos x)(\sin x-3\cos x)$

Solve, for $0°⩽x⩽180°$ the equation, $\frac{\mathrm{d}y}{\mathrm{d}x}=0$

Give your answers to the nearest whole number.

Figure 3 shows a right triangular prism $ABCDEF$. A cross section $ABC$ of the prism is a triangle in which $AB=AC=r\mathrm{cm}$ and $\angle CAB=\frac{\mathit{\pi }}{\mathit{3}}$radians.

In the prism

$AE=BF=CD=5\text{ cm}ED=EF=r\text{ cm}$ and $\angle DEF=\frac{\mathit{\pi }}{\mathit{3}}$ radians.

Show that the volume of the prism is $\frac{5\sqrt{3}}{4} r^2{\text{cm}}^3$

Figure 3 shows a right triangular prism $ABCDEF$. A cross section $ABC$ of the prism is a triangle in which $AB=AC=r\mathrm{cm}$ and $\angle CAB=\frac{\mathit{\pi }}{\mathit{3}}$radians.

The volume of the prism is increasing in such a way that the size of $\angle CAB$ and the size of $\angle DEF$ remain constant and the length of $AE$, the length of $BF$ and the length of $CD$ remain constant.
The lengths of $AB,AC,ED$ and $EF$ are each increasing at a constant rate of 0.2cm / s

Find the exact rate of increase, in cm³ / s, of the volume of the prism when the area of the rectangular face $BCDF$ is 60 cm²

Using formulae (opens in a new tab) , show that

(i) $\sin 2A=2\sin A\cos A$

(ii) $\cos 2A=2{\cos }^{2}A-1$

$f\left(\theta \right)=\frac{2\tan \theta }{1+{\tan }^2\theta }$

Show that $\mathrm{f}\left(\theta \right)=\sin 2\theta$

Solve, in radians to 3 significant figures, for $-\frac{\pi }{2}⩽x⩽\frac{\pi }{2}$, the equation

$5\tan \left(x+\frac{\pi }{6}\right)=\left[1+{\tan }^2\left(x+\frac{\pi }{6}\right)\right]\left[1-2{\cos }^2\left(x+\frac{\pi }{6}\right)\right]$

Using calculus, find the exact value of

${\int }_0^{\frac{\pi }{2}}\left(\frac{4\tan \theta }{1+{\tan }^2\theta }-\cos 5\theta +2\right) \mathrm{d}\theta$

In triangle $ABC$, $AB=3x\mathrm{cm}$, $BC=5x\mathrm{cm}$ and $\angle ABC=110°$

Find, in degrees to one decimal place, the size of $\angle BCA$

In triangle $ABC$, $AB=3x\mathrm{cm}$, $BC=5x\mathrm{cm}$ and $\angle ABC=110°$

The area of triangle $ABC$ is 24 cm²

Find, to 3 significant figures, the value of $x$

Figure 1 shows triangle $XYZ$ in which

$XY=(x+2)\text{ cm}XZ=(2x+4)\text{ cm}YZ=(2x-1)\text{ cm}\mathrm{and}\angle YXZ=60°$

Find the value of $x$

Give your answer in the form $p+q\sqrt{3}$ where $p$ and $q$ are integers to be found.

Using the formulae (opens in a new tab), show that

(i) $\sin 2\theta =2\sin \theta \cos \theta$

[2]

(ii) $\cos 2\theta =2{\cos }^2\theta -1$

[3]

Given that $\theta \ne (90°+180°n)$ where $n\in \mathrm{\mathbb{Z}}$

use the results from part (a) to show that $\sin 2\theta -\tan \theta$ can be written as $\tan \theta \cos 2\theta$

Solve for $0<x<360$

$\sin 2x°-\tan x°=0$