Trigonometry (AQA GCSE Further Maths): Exam Questions

$ABCD$ is a rhombus with side length $8\mathrm{cm}$

Angle $ABC=60°$

Work out the area of the rhombus.

Give your answer in the form $a\sqrt{b}$ cm² where $a$ and $b$ are integers.

............................${\mathrm{cm}}^2$

Use the sine rule to work out the size of obtuse angle $x$.

.............................. degrees

Here is a triangle.

Use the cosine rule to work out the ratio $b^2:a^2$

Here is a triangle.

Use the cosine rule to work out the value of $x$.

$SQR$ is a right-angled triangle.

$P$ is a point on $SQ$.
Angle $SPR$ = 45°
$M$ is the midpoint of $QR$.
$k$ is a constant.

Work out the coordinates of $M$.

$ABC$ is a right-angled triangle.

$ACD$ is an isosceles triangle.

All dimensions are in centimetres.

Show that $AC=5x$

Work out an expression, in ${\mathrm{cm}}^2$, for the area of quadrilateral $ABCD$.

Give your answer in the form $p{x}^2$ where $p$ is an integer.

..........................${\mathrm{cm}}^2$

$PQRSTU$ is a triangular prism.

$PQRS$ is a rectangle and angle $QRU=90°$

$PQ=10\mathrm{cm}$ $QR=12\mathrm{cm}$ $UR=7\mathrm{cm}$     

$M$ is the midpoint of $PQ$.

Calculate the size of the angle between the line $UM$ and the plane $PQRS$.

 ........................ degrees

Calculate the size of the angle between the planes $UMR$ and $UQR$.

........................... degrees

In triangle $PQR$, $\cos P=\frac{1}{3}$

Show that triangle $PQR$is isosceles.

$VABCD$ is a pyramid with a horizontal rectangular base $ABCD$.

$V$ is directly above the centre of the base.

$VA=VB=VC=VD=10\mathrm{cm}$

$AB=8\mathrm{cm}$ $BC=6\mathrm{cm}$

$M$ is the midpoint of $BC$.

Work out the size of angle $VMD$.

............................ degrees

$ABCDEFGH$ is a cube with side length $32\mathrm{cm}$

$M$ and $N$ are points on $DH$ and $CG$ respectively.

Work out the size of the angle that the line $BM$ makes with the plane $ABCD$.

...................degrees

Angle $y$ is acute.

$\tan y=\frac{p+1}{p-1}$ where $p$ is a constant greater than $1$

Work out the expression for $\sin y$

Give your answer in the form $\frac{ap+b}{\sqrt{c{p}^2+d}}$ where $a,b,c$ and $d$ are integers.

You may use a diagram to help you.

Pyramid $VABCD$ has a horizontal rectangular base.

$X$ is the centre of the base.
$V$ is vertically above $X$.

$VB=VC=17\mathrm{cm}$     $AB=22\mathrm{cm}$     $BC=16\mathrm{cm}$

Work out the angle between the planes $VBC$ and $ABCD$.

.............................degrees