IB Maths AI HL Sl 3 3 Applications of Trigonometry Questions

Practise solving triangle, elevation, depression and bearing problems by defining directions consistently and interpreting lengths or angles with units.

[Maximum number: 7]

The following question compares the distance and direction between cities on a flat surface to the distance and direction between cities on a sphere.
Consider a model where the cities of Bogotá, Moscow, and Nairobi lie on a flat surface. In this model, Nairobi is 6000 km due south of Moscow and Bogotá is 12500 km due west of Nairobi, as shown in the following diagram.

Figure for Question IB Maths AI HL Sl 3 3 Applications of Trigonometry Questions question 1 — IB Maths AI HL

(a)

Find the distance from Bogotá to Moscow.

[ 2 ]

attempt to use Pythagoras' theorem

\begin{aligned} & \sqrt{12500^{2}+6000^{2}} \\ & =13900(\mathrm{~km}) \end{aligned}

Note: Answers to more than 3 sf are not appropriate and should be penalised. A1 will be given only for 13900 km ( 3 sf ) or 14000 km (nearest 500 km ).

(b)

Find the bearing of Moscow from Bogotá. Give your answer in degrees.

In reality, these three cities lie on the curved surface of the Earth which will change the distances and directions found in part (a).

Now consider a curved model using a coordinate system (x, y, z) with its origin, O , at the centre of the Earth. The units of this system are thousands of kilometres and the Earth is modelled as a sphere with radius 6000 km . The North Pole, P, lies on the z-axis, and Nairobi, N, is modelled as being on the equator and lying on the y-axis.

Figure for Question (b) — IB Maths AI HL

P has position vector $\overrightarrow{\mathrm{OP}}=\boldsymbol{p}=\left(\begin{array}{l}0 \\ 0 \\ 6\end{array}\right)$ and N has position vector $\overrightarrow{\mathrm{ON}}=\boldsymbol{n}=\left(\begin{array}{l}0 \\ 6 \\ 0\end{array}\right)$ .

[ 3 ]

attempt to use an appropriate inverse trig ratio
e.g. $\arctan \left(\frac{12500}{6000}\right)$ or $\arctan \left(\frac{6000}{12500}\right)$
identifying the correct angle
$=(0) 64.4^{\circ}$ OR $064^{\circ}(64.3589 \ldots)$

Note: The two method marks can be awarded independently. The first can come from any geometry leading to an inverse trigonometric ratio. The second requires recognizing which angle is the bearing, possibly on a diagram. 64.4 appearing in intermediate working is not sufficient.

(c)

Show that $\theta=57.3^{\circ}$ , correct to three significant figures.

Bogotá, B , is west of Nairobi and has position vector $\overrightarrow{\mathrm{OB}}=\boldsymbol{b}=\left(\begin{array}{c}6 \sin 120^{\circ} \\ 6 \cos 120^{\circ} \\ 0\end{array}\right)$ .

[ 2 ]

attempt to use $\frac{\theta}{360} \times 2 \pi r=6$ OR $s=r \theta$

\frac{\theta}{360} \times 2 \pi \times 6=6 \quad \text { OR } \quad 6=6 \theta \Rightarrow \theta=1 \text { radian }

$\theta=57.2957 \ldots{ }^{\circ}$

Note: The fourth significant figure must be seen somewhere to award the A1.

$\theta=57.3^{\circ}$ to 3 significant figures

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IB Maths AI HL Sl 3 3 Applications of Trigonometry Questions

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IB Maths AI HL Sl 3 3 Applications of Trigonometry Questions question 1

Question (a)

Question (b)

Question (c)