Trigonometric Ratios (Edexcel IGCSE Further Pure Maths): Exam Questions

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 $ABCD$

Figure 1 shows a right prism $ABCDEF$
Triangle $CDE$is a cross section of the prism.

• $\angle DCE=\angle ABF=90°$

• $\angle EDC=\angle FAB=40°$

• $DE=AF=15\mathrm{cm}$

• $EF=DA=CB=20\mathrm{cm}$

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

Figure 4 shows a right pyramid with vertex $V$ and base $ABCDE$which is a regular pentagon.

$$\begin{gathered} AB=BC=CD=DE=EA=2x\mathrm{cm} \\ VA=VB=VC=VD=VE=3x\mathrm{cm} \end{gathered}$$

Find, in degrees to one decimal place, the size of the angle between the plane $VBC$ and the base $ABCDE$

Figure 3 shows a right pyramid with a horizontal square base.

$$\begin{gathered} AB=BC=CD=DA=x\mathrm{cm} \\ AV\mathit{=}BV\mathit{=}CV\mathit{=}DV\mathit{=}x\mathrm{cm} \end{gathered}$$

$O$ is the point of intersection of the diagonals of the base.
The vertex $V$ of the pyramid is vertically above $O$

Show that $VO=\frac{\sqrt{2}}{2}x\mathrm{cm}$

Find, in degrees, the size of the angle $AVC$

Find, in degrees to one decimal place, the size of the angle between the plane $VAB$ and the plane $VAD$

The volume of the pyramid is 200 cm³

Given that the $\mathrm{volume}\mathrm{of}\mathrm{a}\mathrm{pyramid}=\frac{1}{3}\times \mathrm{base}\mathrm{area}\times \mathrm{height}$

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

The point $A$ has coordinates (–5, 3), the point $B$ has coordinates (4, 0) and the point $C$ has coordinates (–1, 5).

The line $l$ passes through $C$ and is perpendicular to $AB$.

Find an equation of $l$ .
Give your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

The line $l$ intersects $AB$ at the point $D$.

Show that the coordinates of $D$ are (-2, 2).

The point $A$ has coordinates (–5, 3), the point $B$ has coordinates (4, 0) and the point $C$ has coordinates (–1, 5).

Show that $l$ is not the perpendicular bisector of $AB$.

Find the value of $\tan \angle ABC$.
Give your answer in its simplest form.

Figure 2 shows a right pyramid with a square base $ABCD$ and vertex $E$.

The base of the pyramid is horizontal with $AB=BC=12\mathrm{cm}$.
The diagonals of the base intersect at the point $O$.

The vertex $E$ of the pyramid is vertically above $O$ and the angle between $EA$and the plane $ABCD$ is 30°

The height of the pyramid is $h$ cm.

Find the exact value of $h$

The point $F$ lies on $AD$ such that $AF:FD=1.4$

Calculate, to the nearest degree, the size of the angle $EFO$.