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

Figure 1 shows a shape $ABC$in which $AOB$ is a triangle, $AOC$ is a straight line and $OBC$ is a sector of a circle with centre $O$.

$AO=10$cm $,OC=OB=6$cm and the length of arc $BC=π$ cm.

Find, to 3 significant figures, the length of $AB$.

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 $\mathrm{cos}(A-B)--\mathrm{cos}(A+B)=2\mathrm{sin}A\mathrm{sin}B$

Hence express $2\mathrm{sin}5x\mathrm{sin}3x$ in the form $\mathrm{cos}mx-\mathrm{cos}nx$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 integral _ 0 ^ pi / 6 end ^ 4 sin 5 theta sin 3 theta 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.

$AF=DEBF=CEAD=FE=BCAB=DC=24$ cm

$\text{angle} A B F = \text{angle} D C E = 45 ^{\circ} \text{angle} B A F = \text{angle} C D E = 60 ^{\circ}$

Using a formula from page 2,

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

Without using a calculator,

show that $B F = 12 ( 3 \sqrt{2} - \sqrt{6} )$cm

The angle between the plane $AEB$and the plane $ABCD$is $65^{\circ}$

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 , show that

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

(ii) $\mathrm{sin}2A=2\mathrm{sin}A\mathrm{cos}A$

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

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

$8 \text{cos} ^{3} ( \frac{ \theta }{2} ) - 6 \text{cos} ( \frac{ \theta }{2} ) - 1 = 0$ for 0 less-than or slanted equal to theta less-than or slanted equal to 2 pi

Use algebraic integration to find the exact value of

integral _ 0 ^ pi / 6 end ^ ( 4 cos ^3 theta - sin 2 theta ) d theta

Triangle $ABC$is such that

$AC=10\mathrm{cm}BC=7\mathrm{cm} \mathrm{angle}CAB=25^{\circ}$

Given that angle $ABC$is obtuse,

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

Using a formula , show that

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

Hence, solve the equation

$\mathrm{tan}A^{\circ}-\mathrm{tan}2A^{\circ}=0$ for 0 less-than or slanted equal to A less-than or slanted equal to 180

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

$\cos ( x - \frac{\pi }{6} ) = \sin x$ for negative pi less-than or slanted equal to x less-than or slanted equal to 2 pi

Figure 2 shows triangle $AOB$

$\overset{\rightarrow}{OA}=4a+5b\overset{\rightarrow}{OB}=8a-b\overset{\rightarrow}{OD}=15a+10b$ where $\vert a \vert = \vert b \vert = 1$

(i) Find $\overset{\rightarrow}{AB}$ in terms of $a$ and $b$

(ii) Find, in its simplest form, the exact value of $\vert \vec{AB} \vert$

Figure 2 shows triangle $AOB$

$\overset{\rightarrow}{OA}=4a+5b\overset{\rightarrow}{OB}=8a-b\overset{\rightarrow}{OD}=15a+10b$ where $\vert a \vert = \vert b \vert = 1$

Find the area of triangle $AOB$

Figure 2 shows triangle $AOB$

$\overset{\rightarrow}{OA}=4a+5b\overset{\rightarrow}{OB}=8a-b\overset{\rightarrow}{OD}=15a+10b$ where $\vert a \vert = \vert b \vert = 1$

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

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

Using a formula, show that

$\mathrm{cos}2θ=2\mathrm{cos}^{2}θ-1$

Using a formula, show that

$\mathrm{cos}2θ=2\mathrm{cos}^{2}θ-1$

Hence show that

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

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

Figure 3 shows part of the curve $C_{1}$ with equation $y=2\mathrm{cos}^{2}θ-1$and part of the curve $C_{2}$ with equation $y=-\mathrm{cos}θ$

Point $B$ is the intersection of $C_{1}$ and $C_{2}$ as shown in Figure 3

Point $A\quad ( \frac{3 \mathrm{\pi } }{4} , 0 )$ is the intersection of $C_{1}$ with the $θ$-axis as shown in Figure 3

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

The finite region $R$, shown shaded in Figure 3, is bounded by the $θ$-axis, $C_{1}$ and $C_{2}$

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

Using the formulae , show that

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

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

Show that

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

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

Solve, for 0 degree less-than or slanted equal to x less-than or slanted equal to 180 degree the equation, $\frac{dy}{dx}=0$

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 C A B equals pi over italic 3radians.

In the prism

$AE=BF=CD=5\text{cm}ED=EF=r\text{cm}$ and angle D E F equals pi over italic 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 C A B equals pi over italic 3radians.

The volume of the prism is increasing in such a way that the size of angle C A B and the size of angle D E F 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 , show that

(i) $\mathrm{sin}2A=2\mathrm{sin}A\mathrm{cos}A$

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

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

Show that $f(θ)=\mathrm{sin}2θ$

Solve, in radians to 3 significant figures, for negative pi over 2 less-than or slanted equal to x less-than or slanted equal to pi over 2, the equation

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

Using calculus, find the exact value of

$\int_{0}^{ \frac{\pi }{2} } ( \frac{4 \tan \\theta }{1 + \tan ^{2} \\theta } - \cos 5 \\theta + 2 )   \mathrm{d} \\theta$

In triangle $ABC$, $AB=3x\mathrm{cm}$, $BC=5x\mathrm{cm}$ and angle A B C equals 110 degree

Find, in degrees to one decimal place, the size of angle B C A

In triangle $ABC$, $AB=3x\mathrm{cm}$, $BC=5x\mathrm{cm}$ and angle A B C equals 110 degree

The area of triangle $ABC$ is 24 cm²

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

Figure 1 shows triangle $XYZ$ in which

X Y = ( x + 2 ) text cm end text X Z = ( 2 x + 4 ) text cm end text Y Z = ( 2 x - 1 ) text cm end text and angle Y X Z = 60 deg

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 , show that

(i) $\mathrm{sin}2θ=2\mathrm{sin}θ\mathrm{cos}θ$

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

Given that $θ\neq(90^{\circ}+180^{\circ}n)$ where $n\inℤ$

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

Solve for $0<x<360$

$\mathrm{sin}2x^{\circ}-\mathrm{tan}x^{\circ}=0$