IB Maths AI SL 3.2 Triangle trigonometry Question Bank

Practise IB Maths AI SL 3.2 by solving non-right triangles and interpreting distances and angles in applied settings.

[Maximum number: 4]

The Voronoi diagram below shows four supermarkets represented by points with coordinates A(0,0), B(6,0), C(0,6) and D(2,2). The vertices X, Y, Z are also shown. All distances are measured in kilometres.

Figure for Question SL 3.2—Triangle trigonometry question 1 — IB Maths AI SL

Given that the exact length of [X Y] is $\sqrt{32}$ , find the size of $X \hat{Y} Z$ in degrees.

METHOD 1 (cosine rule)
length of XZ is $\sqrt{80}(4 \sqrt{5}, 8.94427 \ldots)$

Note: Accept 8.94 and 8.9. attempt to substitute into cosine rule

\cos X \hat{Y} Z=\frac{80+32-80}{2 \times \sqrt{80} \sqrt{32}} \quad(=0.316227 \ldots)

Note: Award A1 for correct substitution of XZ, YZ, $\sqrt{32}$ values in the cos rule. Exact values do not need to be used in the substitution.

(\mathrm{X} \hat{\mathrm{Y}} \mathrm{Z}=) 71.6^{\circ}\left(71.5650 \ldots{ }^{\circ}\right)

Note: Last A1 mark may be lost if prematurely rounded values of XZ, YZ and/or XY are used.METHOD 2 (splitting isosceles triangle in half)
length of XZ is $\sqrt{80}(4 \sqrt{5}, 8.94427 \ldots)$
Note: Accept 8.94 and 8.9.
required angle is $\cos ^{-1}\left(\frac{\sqrt{32}}{2 \sqrt{80}}\right)$
Note: Award A1 for correct substitution of XZ (or YZ), $\frac{\sqrt{32}}{2}$ values in the cos rule. Exact values do not need to be used in the substitution.
$(\mathrm{X} \hat{\mathrm{Y}} \mathrm{Z}=) 71.6^{\circ}\left(71.5650^{\circ}\right)$
Note: Last A1 mark may be lost if prematurely rounded values of XZ, YZ and/or XY are used.

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SL 3.2—Triangle trigonometry question 1