In the diagram, S, T and K lie in the same horizontal plane - NSC Mathematics - Question 7 - 2023 - Paper 2

Using the triangle RKS, we apply the sine rule:

$\frac{RS}{\sin \beta} = \frac{SK}{\sin(90^\circ - \beta)}$

This leads to: $RS \cos \beta = SK \sin \beta$

Substituting our earlier expression for SK:

$RS \cos \beta = \frac{2q}{p \sin \alpha} \sin \beta$

Thus: $RS = \frac{2q \tan \beta}{p \sin \alpha}$.

From the second part, we substitute the known values into the equation:

$70 = \frac{2(2500) \tan(42^\circ)}{80 \sin \alpha}$

Solving for sin α:

1. Calculate $\tan(42^\circ) \approx 0.900$,

2. Substitute values: $70 = \frac{5000 \cdot 0.900}{80 \sin \alpha}$

3. Simplifying this: $70 \sin \alpha = \frac{5000 \cdot 0.900}{80}$

4. Finally, calculating sin α: $\sin \alpha \approx 0.8$

This gives: $\alpha \approx 53.51^\circ$.