In die diagram is A, B en C punte in dieselfde horizontale vlak - NSC Mathematics - Question 7 - 2023 - Paper 2

To calculate AB, we use the cosine rule again in triangle ACB:

$AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(100°)$

Substituting in the values we found and known:

$AB^2 = (10\sqrt{3})^2 + 8^2 - 2 \cdot (10\sqrt{3}) \cdot 8 \cdot \cos(100°)$

Calculating this gives:

$AB^2 = 300 + 64 - 2 \cdot 10\sqrt{3} \cdot 8 \cdot (-0.1736)$

After evaluating, we find:

$AB \approx 20.30 \text{ units}$

To find the angle ADB, we can use the sine rule in triangle ADB:

$\frac{\sin(ADB)}{AB} = \frac{\sin(73.4°)}{AD}$

Substituting in the known values:

$\frac{\sin(ADB)}{20.30} = \frac{\sin(73.4°)}{20}$

Solving this gives:

$\sin(ADB) = \frac{20.30 \cdot \sin(73.4°)}{20}$

Calculating this value, we find:

$ADB \approx 76.58°$