IB Maths AA SL 3.1 Geometry and Trigonometry Sl Content Questions

Practise IB Mathematics AA SL 3.1 by solving triangle, radian, coordinate-geometry and trigonometric-graph problems with clear diagrams.

Exam points

Solve 3D length, midpoint, volume or surface-area problems by combining coordinates, solid geometry and right-triangle distances.
Apply right/non-right triangle trigonometry, Pythagoras, bearings and elevation/depression to distances, angles, areas or travel models.
Convert and use radians to calculate arc lengths, sector/segment areas and related geometric parameters.
Use the unit circle, exact values, identities and double-angle relationships to transform or simplify trigonometric expressions.
Graph and analyse sine, cosine or tangent functions, recover amplitude/period/phase parameters, and solve finite-interval equations or periodic models.

Covered syllabus points

SL 3.1—Three-dimensional geometry SL 3.2—Triangle trigonometry SL 3.3—Applications of trigonometry SL 3.4—Radian measure SL 3.5—Unit circle and exact values SL 3.6—Trigonometric identities SL 3.7—Trigonometric functions SL 3.8—Trigonometric equations

[Maximum number: 8]

All lengths in this question are in centimetres.
A solid metal ornament is in the shape of a right pyramid, with vertex V and square base ABCD . The centre of the base is X . Point V has coordinates (1,5,0) and point A has coordinates (-1,1,6).

(a)

Find AV .

[ 2 ]

attempt to use the distance formula to find AV

\begin{aligned} & \sqrt{(1-(-1))^{2}+(5-1)^{2}+(0-6)^{2}} \\ & =7.48331 \ldots \\ & =7.48(\mathrm{~cm})(=\sqrt{56} \text { or } 2 \sqrt{14}) \end{aligned}

(b)

Given that $\mathrm{A} \hat{\mathrm{V}} \mathrm{B}=40^{\circ}$ , find AB .

The volume of the pyramid is $57.2 \mathrm{~cm}^{3}$ , correct to three significant figures.

[ 3 ]

(b) METHOD 1 attempt to apply cosine rule OR sine rule to find AB
$(\mathrm{AB}=) \sqrt{7.48 \ldots^{2}+7.48 \ldots^{2}-2 \times 7.48 \ldots \times 7.48 \ldots \cos \left(40^{\circ}\right)}$ OR $\frac{\mathrm{AB}}{\sin 40^{\circ}}=\frac{\sqrt{56}}{\sin 70^{\circ}}$ $=5.11888 \ldots$ $=5.12(\mathrm{~cm})$
METHOD 2
Let M be the midpoint of $[\mathrm{AB}]$
attempt to apply right-angled trigonometry on triangle AVM
$=2 \times 7.48 \ldots \times \sin \left(20^{\circ}\right)$ $=5.11888 \ldots$ $=5.12(\mathrm{~cm})$

(c)

Find the height of the pyramid, VX.

A second ornament is in the shape of a cuboid with a rectangular base of length $2 x \mathrm{~cm}$ , width $x \mathrm{~cm}$ and height $y \mathrm{~cm}$ . The cuboid has the same volume as the pyramid.

[ 3 ]

METHOD 1 equating volume of pyramid formula to 57.2
$\frac{1}{3} \times 5.11 \ldots{ }^{2} \times h=57.2$
(A1)
$h=6.54886 \ldots$ $h=6.55(\mathrm{~cm})$
METHOD 2
Let M be the midpoint of AB ]
$A V^{2}=A M^{2}+M X^{2}+X V^{2}$ $\Rightarrow X V=\sqrt{7.48 \ldots^{2}-\left(\frac{5.11 \ldots}{2}\right)^{2}-\left(\frac{5.11 \ldots}{2}\right)^{2}}$ $h=6.54886 \ldots$ $h=6.55(\mathrm{~cm})$

[Maximum number: 5]

A monument is in the shape of a right cone with a vertical height of 20 metres. Oliver stands 5 metres from the base of the monument. His eye level is 1.8 metres above the ground and the angle of elevation from Oliver's eye level to the vertex of the cone is $58^{\circ}$ , as shown on the following diagram.

(a)

Find the radius of the base of the cone.

[ 3 ]

attempt to use trigonometry to find the radius of the cone OR Oliver's distance from
centre (r+5)

\begin{aligned} & \tan 58^{\circ}=\frac{18.2}{r+5} \text { OR } \frac{r+5}{\sin 32^{\circ}}=\frac{18.2}{\sin 58^{\circ}} \text { OR }(r+5=) 11.3726 \ldots \\ & r=6.37262 \ldots(\mathrm{~m}) \\ & (r=) 6.37(\mathrm{~m}) \end{aligned}

(b)

Find the volume of the monument.

[ 2 ]

attempt to substitute h=20 and their radius into the correct volume of cone formula

\begin{aligned} & V=\frac{\pi(6.37262 \ldots)^{2}(20)}{3} \\ & =850.540 \ldots \\ & =851\left(\mathrm{~m}^{3}\right) \end{aligned}

Note: Accept $849.840 \ldots(850)$ obtained from using r=6.37.

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)